2.1 Classical annular Khovanov homology

This section reviews the construction of sutured annular Khovanov homology [Reference Asaeda, Przytycki and SikoraAPS04, Reference Bar-NatanBar05, Reference RobertsRob13]. We will refer to it as classical annular homology, to distinguish it from the quantum version discussed in § 2.2. Let $I:= [0,1]$ denote the unit interval; we fix the notation $\mathbb {A}$ for the annulus $S^1\times I$. An annular link is a link in the thickened annulus $\mathbb {A}\times I$, and its diagram is a projection onto the first factor of $\mathbb {A} \times I$. Link diagrams are disjoint from the boundary of $\mathbb {A}$. Identifying $S^1\times (0,1)$ with $\mathbb {R}^2$ minus a point, we represent the annulus by simply indicating the deleted point using the symbol $\times$. Figure 1 illustrates an example of a link diagram.

Figure 1. An annular link diagram.

Let $\mathcal {BN}(\mathbb {A})$ denote the Bar-Natan category of the annulus [Reference Bar-NatanBar05]. Its objects are formal $\mathbb {Z}$-linear combinations of formally graded collections of simple closed curves in $\mathbb {A}$. Morphisms are matrices whose entries are formal $\mathbb {Z}$-linear combinations of dotted cobordisms embedded in $\mathbb {A}\times I$, modulo isotopy relative to the boundary, subject to the Bar-Natan relations, Figure 2.

Figure 2. Bar-Natan relations.

Let $D$ be a diagram for an oriented annular link $L$. We briefly review the construction of the chain complex $[[D]]$; a complete treatment can be found in [Reference Bar-NatanBar05]. To begin, one first forms the cube of resolutions as follows. Label the crossings of the diagram by $1,\ldots , n$. Every crossing may be resolved in two ways, called the 0-smoothing and the 1-smoothing, as in (

). For each $u = (u_1,\ldots , u_n)\in \{0,1\}^n$, perform the $u_i$-smoothing at the $i$th crossing. The resulting diagram is a collection of disjoint simple closed curves in $\mathbb {A}$, which we denote $D_u$. Thinking of elements of $\{0,1\}^n$ as vertices of an $n$-dimensional cube, decorate the vertex $u$ by the smoothing $D_u$.

(2.1)

Let $v=(v_1,\ldots , v_n)$ and $u=(u_1,\ldots , u_n)$ be vertices which differ only in the $i$th entry, where $v_i = 0$ and $u_i=1$. Then the diagrams $D_v$ and $D_u$ are the same outside of a small disk around the $i$th crossing. There is a cobordism from $D_v$ to $D_u$, which is the obvious saddle near the $i$th crossing and the identity (product cobordism) elsewhere. We will call this the saddle cobordism from $D_v$ to $D_u$ and denote it by $d_{v,u}$. Decorate each edge of the $n$-dimensional cube by these saddle cobordisms. We now have a commutative cube in the category $\mathcal {BN}(\mathbb {A})$. There is a way to assign $s_{u,v} \in \{0,1\}$ to each edge so that multiplying the edge map $d_{v,u}$ by $(-1)^{s_{v,u}}$ results in an anti-commutative cube (see [Reference Bar-NatanBar05, § 2.7]; see also [Reference Lipshitz and SarkarLS14a, Definition 4.5]).

For $u=(u_1,\ldots , u_n) \in \{0,1\}^n$, let $\vert u \vert = \sum _i u_i$. Now form the chain complex $[[D]]$ by setting

\[ [[D]]^i = \bigoplus_{\vert u \vert = i+ n_-} D_{u}\{i + n_+-n_-\}, \]

where $n_-$ and $n_+$ are the numbers of negative and positive crossings in $D$, and the brackets $\{-\}$ denote the formal grading shift in $\mathcal {BN}(\mathbb {A})$. The differential is given on each summand by the edge map $(-1)^{s_{v,u}}d_{v,u}$. Anti-commutativity of the cube ensures that $[[D]]$ is a complex.

To obtain classical annular Khovanov homology, one applies the annular TQFT

\[ \mathcal{F}_\mathbb{A} : \mathcal{BN}(\mathbb{A}) \to \text{Mod}(\mathbb{Z}) \]

defined as follows. Let $V_\mathbb {Z}$ and $W_\mathbb {Z}$ be free rank two $\mathbb {Z}$-modules with bases $v_-, v_+$ and $w_-, w_+$, respectively. Equip each with two gradings, the quantum grading $\text {qdeg}$ and the annular grading $\text {adeg}$, defined on generators by

(2.2)\begin{align} & \text{qdeg}(v_\pm) = 0, \quad \enspace\, \text{qdeg}(w_\pm) ={\pm}{1}, \end{align}

(2.3)\begin{align} & \text{adeg}(v_\pm) \,={\pm}{1}, \quad \text{adeg}(w_\pm)= 0.\hphantom{0\,\,}\end{align}

We follow the grading convention of [Reference Beliakova, Putyra and WehrliBPW19]; note that the quantum grading on $V$ is different than the quantum grading appearing elsewhere in the literature; see Remark 2.3.

There are two types of simple closed curves in $\mathbb {A}$; essential curves and trivial curves which bound disks in $\mathbb {A}$. The functor $\mathcal {F}_\mathbb {A}$ assigns $V_\mathbb {Z}$ to each essential circle and $W_\mathbb {Z}$ to each trivial circle. Then for $\mathcal {C}\subset \mathbb {A}$ a collection of disjoint simple closed curves with $e$ essential and $t$ trivial circles, the free abelian group $\mathcal {F}_\mathbb {A}(\mathcal {C}) =$ $\otimes ^e V_\mathbb {Z} \otimes ^t W_\mathbb {Z}$ (where the tensor product is taken over $\mathbb {Z}$) has a standard basis consisting of a label of $v_-$ or $v_+$ on each essential circle, and $w_-$ or $w_+$ on each trivial one. Following the conventions in [Reference Beliakova, Putyra and WehrliBPW19], a generator of $\mathcal {C}$ will be represented as a choice of counterclockwise or clockwise orientations on each essential circle, corresponding to $v_+$ and $v_-$, respectively, and either a dot or no dot on each trivial circle, corresponding to $w_-$ and $w_+$. We will often switch between the diagrammatic and algebraic representations of generators, Figure 3.

Figure 3. Diagrammatic representation of generators.

To define $\mathcal {F}_\mathbb {A}$ on a cobordism, it is enough to consider cups, caps, and saddles. To a cup, $\mathcal {F}_\mathbb {A}$ assigns the unit $\varepsilon : \mathbb {Z} \to W_\mathbb {Z}$ defined by $\varepsilon (1) = w_+$. To a cap, $\mathcal {F}_\mathbb {A}$ assigns the counit $\eta : W_\mathbb {Z} \to \mathbb {Z}$ defined by

\begin{align*}&\eta(w_-) = 1, \quad \eta(w_+) = 0. \end{align*}

A saddle is assigned one of the six maps shown in Figure 4, depending on whether it is a merge or a split and the types of curves involved.

Definition 2.1 If $D$ is a diagram for an annular link $L$, define the annular Khovanov complex of $D$ to be

\[ CKh_{\mathbb{A}}(D):= \mathcal{F}_\mathbb{A}([[D]]); \]

it is an invariant of $L$ up to chain homotopy equivalence.

Figure 4. Surgery formulas in classical annular Khovanov homology. The table lists topological types of surgeries and corresponding maps on generators.

Remark 2.2 Some of these formulas have an interpretation in terms of relations on cobordisms, as follows. Let $\mathcal {BBN}(\mathbb {A})$ [Reference Beliakova, Putyra and WehrliBPW19] denote the quotient of $\mathcal {BN}(\mathbb {A})$ by Boerner's relation, introduced in [Reference BoernerBoe08], which says that any cobordism carrying a dot and an essential curve is set to $0$ (see Figure 5).

Figure 5. Boerner's relation.

Algebraically, a dot on a cobordism corresponds to multiplication with $w_-$. Then, for a trivial circle $C$, the standard generator $w_+$ (respectively $w_-$) of $\mathcal {F}_\mathbb {A}(C) = W_\mathbb {Z}$ is the image of $1\in \mathbb {Z}$ under the undotted (respectively dotted) cup cobordism from $\varnothing$ to $C$. The surgery formulas of Figure 4 then imply that $\mathcal {F}_\mathbb {A}$ factors through $\mathcal {BBN}(\mathbb {A})$. Algebraically, Boerner's relation can be seen as enforcing the equations $w_-\cdot v_- = w_- \cdot v_+ = 0$.

2.2 Overview of quantum annular homology

This section outlines the construction of the Beliakova–Putyra–Wehrli quantum annular link homology [Reference Beliakova, Putyra and WehrliBPW19]. The theory is built over a commutative ring $\Bbbk$ and a unit $\mathfrak {q}\in \Bbbk$. We set $\Bbbk :=\mathbb {Z}[\mathfrak {q},\mathfrak {q}^{-1}]$; the distinguished unit $\mathfrak {q}\in \Bbbk$ is the same $\mathfrak {q}$ appearing in $\mathbb {Z}[\mathfrak {q},\mathfrak {q}^{-1}]$. The main object is the quantum annular TQFT

\[ \mathcal{F}_{\mathbb{A}_{\mathfrak{q}}} : \mathcal{BN}_{\mathfrak{q}}(\mathbb{A}) \to \text{Mod}(\Bbbk), \]

where $\text {Mod}(\Bbbk )$ is the category of graded $\Bbbk$-modules and $\mathcal {BN}_{\mathfrak {q}}(\mathbb {A})$ is a certain deformation of the Bar-Natan category of the annulus. We will give a brief overview of the functor $\mathcal {F}_{\mathbb {A}_\mathfrak {q}}$ and state a main theorem [Reference Beliakova, Putyra and WehrliBPW19, Theorem 6.3].

Let $\mathcal {BN}(n,m)$ denote the Bar-Natan category of the rectangle with $n$ points on the bottom and $m$ on the top. Its objects are formal direct sums of formally graded planar tangles in $I^2$ with $n$ endpoints on $I \times \{0\}$ and $m$ endpoints on $I \times \{1\}$, cf. Figure 6. Such a tangle will be called a planar $(n,m)$-tangle. Morphisms in $\mathcal {BN}(n,m)$ are matrices whose entries are formal $\Bbbk$-linear combinations of embedded dotted cobordisms in $I^3$ between planar $(n,m)$-tangles, subject to the Bar-Natan relations (see Figure 2).

Figure 6. A planar $(3,1)$-tangle.

A seam of $\mathbb {A}=S^1\times I$, denoted $\mu$, is an interval $\{*\}\times I$. In our representation of the interior of the annulus as $\mathbb {R}^2\smallsetminus \times$, we will fix the seam as the positive $x$-axis, ending on the left in $\times$. See Figure 8 for an example.

The quantum Bar-Natan category of the annulus, denoted $\mathcal {BN}_{\mathfrak {q}}(\mathbb {A})$, is a deformation of $\mathcal {BN}(\mathbb {A}$). The objects of $\mathcal {BN}_{\mathfrak {q}}(\mathbb {A})$ are nearly the same as those of $\mathcal {BN}(\mathbb {A})$, with the slight modification that curves in $\mathbb {A}$ must be transverse to $\mu$. Morphisms in $\mathcal {BN}_{\mathfrak {q}}(\mathbb {A})$ are also similar to those in $\mathcal {BN}(\mathbb {A})$. In $\mathcal {BN}_{\mathfrak {q}}(\mathbb {A})$, isotopic cobordisms are identified if the isotopy fixes the membrane $\mu \times I \subset \mathbb {A}\times I$. Otherwise, the cobordisms are scaled according to the degree of the part of the cobordism that passes through the membrane during the isotopy, accounting also for the co-orientation of the membrane induced by the standard orientation of the core circle of $\mathbb {A}$. These will be referred to as trace moves. The relations are depicted in Figure 7; for details, see [Reference Beliakova, Putyra and WehrliBPW19, § 6.2].

Figure 7. Relations in $\mathcal {BN}_{\mathfrak {q}}(\mathbb {A})$.

The Bar-Natan relations (Figure 2) are imposed, where the local pictures are understood to be disjoint from the membrane.

By general position, if two annular cobordisms are isotopic, then they are related by a sequence of trace moves and isotopies fixing the membrane. Therefore, if two cobordisms $S, S'\subset \mathbb {A}\times I$ are isotopic, then $S = \mathfrak {q}^k S'$ as morphisms in $\mathcal {BN}_{\mathfrak {q}}(\mathbb {A})$ for some $k\in \mathbb {Z}$. (See also [Reference Beliakova, Putyra and WehrliBPW19, Proposition 6.2].)

A configuration $\mathcal {C}$ is a collection of disjoint simple closed curves in $\mathbb {A}$ which are transverse to $\mu$. Note that an object of $\mathcal {BN}_{\mathfrak {q}}(\mathbb {A})$ is a formal direct sum of formally graded configurations. Given a configuration $\mathcal {C}$ which intersects $\mu$ in $n$ points, we can cut along $\mu$ to obtain a planar $(n,n)$-tangle $\mathcal {C}^\textrm {cut}$. See Figure 8 for an example. A construction of Chen and Khovanov [Reference Chen and KhovanovCK14] yields graded $\Bbbk$-algebras $A^k$ for each $k\geq 0$ and a functor

\[ \mathcal{F}_{CK} : \mathcal{BN}(n,m) \to \text{gBimod}(A^n, A^m), \]

where $\text {gBimod}(A^n, A^m)$ is the category of graded $(A^n, A^m)$-bimodules. Let $\mathbb {I}^n$ denote the planar tangle consisting of $n$ vertical strands. Then, by definition of $\mathcal {F}_{CK}$, we have $\mathcal {F}_{CK}(\mathbb {I}^n) = A^n$.

Figure 8. Cutting open a configuration $\mathcal {C}$ along $\mu$ to obtain a $(3,3)$-tangle $\mathcal {C}^\textrm {cut}$.

The quantum Hochschild homology, denoted $qHH$ and defined in [Reference Beliakova, Putyra and WehrliBPW19, § 3.8.5], is a deformation of the usual Hochschild homology of bimodules. It takes as input a graded $\Bbbk$-algebra $B$ and a graded $(B,B)$-bimodule $M$. The output $qHH(B,M) = {\bigoplus} _{i\geq 0} qHH_i(B,M)$ is a $\Bbbk$-module. Due to [Reference Beliakova, Putyra and WehrliBPW19, Proposition 6.6] (stating that $qHH_i(\mathcal {F}_{CK}(\mathcal {C}^\textrm {cut})) = 0$ for $i>0$), we will mostly be interested in $qHH_0$. It follows immediately from the definition of $qHH$ that

(2.4)\begin{equation} qHH_0(B,M) = M/ \text{span}_{\Bbbk} \{bm - \mathfrak{q}^{\vert b \vert } mb \mid b\in B, m\in M\}, \end{equation}

where $\vert b \vert$ denotes the degree of $b$.

We are now ready to define $\mathcal {F}_{\mathbb {A}_{\mathfrak {q}}}$ on objects. Let $\mathcal {C}$ be a configuration which intersects $\mu$ in $n$ points. Using the Chen–Khovanov functor, form the $(A^n, A^n$)-bimodule $\mathcal {F}_{CK}(\mathcal {C}^\textrm {cut})$. The quantum annular TQFT $\mathcal {F}_{\mathbb {A}_{\mathfrak {q}}}$ is then defined on objects by

\[ \mathcal{F}_{\mathbb{A}_{\mathfrak{q}}}(\mathcal{C}) : = qHH(A^n, \mathcal{F}_{CK}(\mathcal{C}^{\rm cut})). \]

By [Reference Beliakova, Putyra and WehrliBPW19, Proposition 6.6], we have $qHH_i(\mathcal {F}_{CK}(\mathcal {C}^\textrm {cut})) = 0$ for $i>0$. Suppose that $\mathcal {C}$ consists of $n$ essential curves each intersecting the seam once. Then $\mathcal {C}^\textrm {cut} = \mathbb {I}^n$, so

\[ \mathcal{F}_{\mathbb{A}_{\mathfrak{q}}}(\mathcal{C}) = qHH_0(A^n, A^n). \]

Let $A^n_0 \subset A^n$ denote the subalgebra consisting of elements of degree $0$. By [Reference Beliakova, Putyra and WehrliBPW19, Proposition 6.6], the inclusion $A^n_0 \hookrightarrow A^n$ induces an isomorphism $qHH_0(A^n_0, A^n_0) \cong qHH_0(A^n, A^n)$. Moreover, $A^n_0$ is freely generated over $\Bbbk$ by $2^n$ elements $x_1,\ldots , x_{2^n}$, which are the primitive idempotents of [Reference Beliakova, Putyra and WehrliBPW19, § 5.5]. They are in bijection with the cup diagrams and satisfy $x_ix_j =\delta _{ij} x_i$. It follows from (2.4) that

\[ qHH_0(A^n_0,A^n_0) \cong \Bbbk^{2^n}. \]

Every configuration $\mathcal {C}$ is isomorphic in $\mathcal {BN}_{\mathfrak {q}}(\mathbb {A})$ to a configuration $\mathcal {C}^\circ$ in which every curve intersects the seam at most once. If $\mathcal {C}$ has $e$ essential and $t$ trivial circles, then, by delooping, one obtains

\[ \mathcal{F}_{\mathbb{A}_{\mathfrak{q}}}(\mathcal{C}) \cong \mathcal{F}_{\mathbb{A}_{\mathfrak{q}}}(\mathcal{C}^\circ) \cong \Bbbk^{2^{e+t}}. \]

We have so far only explained what $\mathcal {F}_{\mathbb {A}_{\mathfrak {q}}}$ does on objects. The full construction of $\mathcal {F}_{\mathbb {A}_{\mathfrak {q}}}$ in [Reference Beliakova, Putyra and WehrliBPW19] follows from a more general theory of (twisted) horizontal traces of bicategories, which we will not describe. The definition of $\mathcal {F}_{\mathbb {A}_{\mathfrak {q}}}$ on morphisms follows from this general theory. The rest of this subsection describes the set-up for [Reference Beliakova, Putyra and WehrliBPW19, Theorem 6.3], which is stated as our Theorem 2.5, and which is the main computational tool.

Let $\mathcal {BBN}_{\mathfrak {q}}(\mathbb {A})$ denote the quotient of $\mathcal {BN}_{\mathfrak {q}}(\mathbb {A})$ by Boerner's relation; see Figure 5. The functor $\mathcal {F}_{\mathbb {A}_{\mathfrak {q}}}$ factors through $\mathcal {BBN}_{\mathfrak {q}}(\mathbb {A})$. Given a diagram $D$ for an annular link $L$ such that $D$ is transverse to $\mu$ and the crossings are disjoint from $\mu$, we form the cube of resolutions $[[D]]$ in the usual manner and view the result as a chain complex over the quantized category $\mathcal {BBN}_{\mathfrak {q}}(\mathbb {A})$.

Definition 2.2 If $D$ is a diagram for an annular link $L$ which is transverse to the seam, define the quantum annular Khovanov complex of $D$ to be

\[ CKh_{\mathbb{A}_\mathfrak{q}}(D):= \mathcal{F}_{\mathbb{A}_\mathfrak{q}}([[D]]). \]

The chain complex $CKh_{\mathbb {A}_\mathfrak {q}}(D)$ is an invariant of $L$ up to chain homotopy equivalence by [Reference Beliakova, Putyra and WehrliBPW19, Proposition 6.8].

Let $TL$ denote the additive closure of the formally graded Temperley–Lieb category [Reference Beliakova, Putyra and WehrliBPW19, Appendix A.1]. Its objects are formal direct sums of formally graded finite collections of points on a line, and morphisms are $\Bbbk$-linear combinations of planar tangles between the points, modulo planar isotopy and the local relation that a circle is set to $\mathfrak {q}+\mathfrak {q}^{-1}$. Composition is given by stacking planar tangles; see Figure 9 for an example. There is a functor $S^1\times (-) : TL \to \mathcal {BN}_{\mathfrak {q}}(\mathbb {A})$, which sends a collection of $n$ points to $n$ essential circles in $\mathbb {A}$, each intersecting $\mu$ once, and sends a planar tangle $T$ to the cobordism $S^1\times T$. The relations in $\mathcal {BN}_{\mathfrak {q}}(\mathbb {A})$ (see Figure 7) imply that a torus wrapping once around the annulus evaluates to $\mathfrak {q}+\mathfrak {q}^{-1}$, so that $S^1 \times (-)$ is well defined.

Figure 9. Composition and relations in $TL$.

Let $\text {gRep}(U_{\mathfrak {q}}(\mathfrak {sl}_2))$ denote the category of graded representations of $U_{\mathfrak {q}}(\mathfrak {sl}_2)$. We follow the conventions established in [Reference Beliakova, Putyra and WehrliBPW19, Appendix A.1] concerning $U_{\mathfrak {q}}(\mathfrak {sl}_2)$; also see § 8 of this paper. There is another functor $\mathcal {F}_{TL} : TL \to \text {gRep}(U_{\mathfrak {q}}(\mathfrak {sl}_2))$, defined as follows. Let $V_1 = \langle v_{-1}, v_1 \rangle$ be the fundamental representation of $U_{\mathfrak {q}}(\mathfrak {sl}_2)$ and $V_1^* = \langle v_{-1}^* , v_1^* \rangle$ its dual. Let $V$ be free over $\Bbbk$ with basis $\{v_+, v_-\}$. Consider two $\Bbbk$-linear isomorphisms $\alpha : V_1 \to V$ and $\beta : V_1^*\to V$ defined by

\begin{align*} & \alpha : v_1 \mapsto v_+, \quad \enspace\ \beta: v_1^* \mapsto v_-, \hphantom{000\,\,\,}\\ & \alpha: v_{{-}1} \mapsto v_-, \quad \ \beta: v_{{-}1}^* \mapsto \mathfrak{q}^{{-}1}v_+. \end{align*}

These equip $V$ with two actions of $U_{\mathfrak {q}}(\mathfrak {sl}_2)$, which are detailed in [Reference Beliakova, Putyra and WehrliBPW19, Appendix A.1]. Note that $\beta ^{-1}\circ \alpha : V_1 \to V_1^*$ is not $U_{\mathfrak {q}}(\mathfrak {sl}_2)$-linear, even though $V_1$ and $V_1^*$ are isomorphic as $U_{\mathfrak {q}}(\mathfrak {sl}_2)$-modules.

The functor $\mathcal {F}_{TL}$ assigns $V^{\otimes n}$ to a collection of $n$ points. Since $\beta ^{-1}\alpha$ is not $U_{\mathfrak {q}}(\mathfrak {sl}_2)$-linear, there is an ambiguity in specifying the $U_{\mathfrak {q}}(\mathfrak {sl}_2)$-module structure on $V^{\otimes n}$. The convention is that the $m$th point is assigned $V_1$ if $m$ is odd and $V_1^*$ if $m$ is even, so that the module assigned to $n$ points is given the $U_{\mathfrak {q}}(\mathfrak {sl}_2)$-action according to the identification

\[ V^{\otimes n} \cong V_1 \otimes V_1^* \otimes V_1\otimes \cdots. \]

To define the value of $\mathcal {F}_{TL}$ on any planar tangle, it suffices to specify its value on caps and cups. For a cap $\cap$, $\mathcal {F}_{TL}$ assigns the evaluation map $ev : V\otimes V \to \Bbbk$, defined by

\begin{align*} & v_+\otimes v_+ \mapsto 0, \quad v_+\otimes v_- \mapsto \mathfrak{q}, \\ & v_-\otimes v_- \mapsto 0, \quad v_-\otimes v_+ \mapsto 1. \end{align*}

On a cup $\cup$, $\mathcal {F}_{TL}$ assigns the coevaluation $coev: \Bbbk \to V\otimes V$, defined by

\[ 1 \mapsto v_+\otimes v_- +\mathfrak{q}^{{-}1} v_-\otimes v_+. \]

The evaluation map is always identified with either $V_1 \otimes V_1^* \to \Bbbk$ or $V_1^*\otimes V_1 \to \Bbbk$, and the coevaluation is identified with either $\Bbbk \to V_1 \otimes V_1^*$ or $\Bbbk \to V_1^* \otimes V_1$. With these identifications, the cap and cup are assigned $U_{\mathfrak {q}}(\mathfrak {sl}_2)$-linear maps by $\mathcal {F}_{TL}$.

We have now explained the functors $\mathcal {F}_{TL} : TL\to \text {gRep}(U_{\mathfrak {q}}(\mathfrak {sl}_2))$ and $S^1\times (-): TL\to \mathcal {BN}_\mathfrak {q}(\mathbb {A})$. To compare $\mathcal {F}_{TL}$ with the composition $\mathcal {F}_{\mathbb {A}_\mathfrak {q}}\circ S^1\times (-)$ in the statement of Theorem 2.5, the value of $\mathcal {F}_{\mathbb {A}_\mathfrak {q}}$ on $n$ essential circles intersecting the seam once needs to be given a $U_{\mathfrak {q}}(\mathfrak {sl}_2)$-module structure. Recall that the Chen–Khovanov functor assigns the $\Bbbk$-algebra $A^n$ to the planar tangle $\mathbb {I}^n$ consisting of $n$ vertical strands. There is a distinguished $\Bbbk$-linear isomorphism

(2.5)\begin{equation} qHH_0(A^n, A^n) \cong V^{\otimes n}, \end{equation}

which we now describe. Recall that the inclusion $A^n_0 \hookrightarrow A_n$ induces an isomorphism on $qHH_0$, and that $qHH_0(A^n, A^n_0)$ has a distinguished $\Bbbk$-basis $\{x_1,\ldots , x_{2^n}\}$ corresponding to cup diagrams. Chen and Khovanov in [Reference Chen and KhovanovCK14, § 6] assigned to each $x_i$ an element $p_i\in V^{\otimes n}$ such that the collection $\{p_i\}$ forms a basis of $V^{\otimes n}$. The isomorphism (2.5) is obtained by composing $qHH_0(A^n, A^n) \cong qHH_0(A^n, A^n_0)$ with the assignment $x_i\mapsto p_i$.

Theorem 2.5 [Reference Beliakova, Putyra and WehrliBPW19, Theorem 6.3]

There is a commuting diagram

with the horizontal functor an equivalence of categories.

This theorem will play an important role in determining the values of the differential on generators in the next section.

2.3 Fixing generators

In the construction of Khovanov homotopy types in [Reference Lipshitz and SarkarLS14a] and [Reference Sarkar, Scaduto and StoffregenSSS20] it is important to have a fixed set of generators for each configuration. Due to the definition of $\mathcal {F}_{\mathbb {A}_\mathfrak {q}}$, the situation is more complicated in quantum annular homology. In this subsection, we explain how to fix generators for a general configuration.

We will say that a configuration $\mathcal {C}$ is standard if every component intersects the seam in at most one point. Every configuration $\mathcal {C}$ is isotopic in $\mathbb {A}$ to a standard configuration, denoted $\mathcal {C}^\circ$, which is unique up to planar isotopy of the cut-open planar tangle. Next we explain in detail how Theorem 2.5 gives a canonical choice of generators for $\mathcal {F}_{\mathbb {A}_{\mathfrak {q}}}(\mathcal {C})$ when $\mathcal {C}$ is standard.

For a configuration $\mathcal {C}$, we will write $\mathcal {C}_\textrm {E}$ to denote the essential circles in $\mathcal {C}$ and $\mathcal {C}_\textrm {T}$ to denote the trivial circles. Figure 10 illustrates these conventions.

Figure 10. (a) A configuration $\mathcal {C}$, (b) its standard form $\mathcal {C}^\circ$, (c) $\mathcal {C}_\textrm {E}$, and (d) $\mathcal {C}_\textrm {T}$.

For a cobordism $S\subset \mathbb {A} \times I$, let $\bar {S}$ denote its reflection in the $I$ coordinate. For a configuration $\mathcal {C}$, we will often suppress the notation $\mathcal {F}_{\mathbb {A}_{\mathfrak {q}}}(\mathcal {C})$ when it is clear from context; that is, $x\in \mathcal {C}$ means $x\in \mathcal {F}_{\mathbb {A}_{\mathfrak {q}}}(\mathcal {C})$. Likewise, for a cobordism $S:\mathcal {C}\to \mathcal {C}'$, we will often write $S(x)$ to mean $\mathcal {F}_{\mathbb {A}_\mathfrak {q}}(S)(x)$, where $\mathcal {F}_{\mathbb {A}_\mathfrak {q}}(S): \mathcal {F}_{\mathbb {A}_{\mathfrak {q}}}(\mathcal {C}) \to \mathcal {F}_{\mathbb {A}_{\mathfrak {q}}}(\mathcal {C}')$ is the induced map. For cobordisms $\mathcal {C} \xrightarrow {S_1} \mathcal {C}'$ and $\mathcal {C}' \xrightarrow {S_2} \mathcal {C}''$, we will write $S_2S_1$ to denote their composition.

Suppose that $C\subset \mathbb {A}$ is a trivial circle and set $n=\vert C \cap \mu \vert$. The circle $C$ bounds an embedded disk $D\subset \mathbb {A}$, and we may push the interior of $D$ down in the $I$ coordinate to obtain a cobordism $\Sigma \subset \mathbb {A}\times I$ from the empty set to $C$ which intersects the membrane in exactly $n/2$ arcs. We refer to $\Sigma$ as the cup cobordism on $C$. Similarly, we may pull $D$ up in the $I$ coordinate to obtain the cap cobordism on C, which is simply $\bar {\Sigma }$.

Let $W$ denote the $\Bbbk$-module assigned by $\mathcal {F}_{\mathbb {A}_{\mathfrak {q}}}$ to a trivial circle $C$ which is disjoint from the seam. The module $W = \langle w_-, w_+\rangle$ is free of rank $2$. The standard generator $w_+$ (respectively $w_-$) is the image of $1\in \Bbbk$ under the undotted (respectively dotted) cup cobordism on $C$. Therefore, we will often identify $w_\pm$ with these cup cobordisms. Diagrammatically, we will signify that a trivial circle $C$ in $\mathcal {C}$ is labelled by $w_-$ by drawing a dot on $C$, as in Figure 3.

Suppose that $\mathcal {C}$ is a standard configuration with $e$ essential circles and $t$ trivial circles. Order the trivial circles in some way, and order the essential circles from the innermost to the outermost. The exact ordering of the trivial circles is irrelevant, but it is important to order the essential circles in this way in light of Theorem 2.5 and the asymmetry of the evaluation and coevaluation maps. We have that

(2.6)\begin{equation} \mathcal{F}_{\mathbb{A}_{\mathfrak{q}}}(\mathcal{C}) = V^{\otimes e} \otimes W^{\otimes t}, \end{equation}

where the tensor products above are understood to be over $\Bbbk$, and the identification of the value of $\mathcal {F}_{\mathbb {A}_\mathfrak {q}}$ on $e$ standard essential circles with $V^{\otimes e}$ is the isomorphism from (2.5). The modules $V$ and $W$ are each bigraded, carrying a quantum grading $\text {q deg}$ and an annular grading $\text {adeg}$. The degrees of generators are as in (2.2) and (2.3). Algebraically, we will write a standard generator as

\[ v_{a_1} \otimes \cdots \otimes v_{a_e} \otimes w_{b_1} \otimes \cdots \otimes w_{b_t}, \]

where each $a_i, b_j \in \{-, +\}$, the $v_{a_i}$ label the essential circles, and the $w_{b_j}$ label the trivial circles. We will often shorten the notation to $v_\mathcal {I} \otimes w_\mathcal {J}$, where $\mathcal {I}$ is a sequence of $\pm$ labelling the essential circles and $\mathcal {J}$ is a sequence of $\pm$ labelling the trivial ones. Note also that each standard generator $x = v_\mathcal {I} \otimes w_\mathcal {J} \in \mathcal {C}$ of a standard configuration $\mathcal {C}$ is the image of $v_\mathcal {I}$ under the cobordism

\[ \Sigma_\mathcal{J} : \mathcal{C}_{\rm E} \to \mathcal{C}, \]

which is the identity on $\mathcal {C}_\textrm {E}$ and a cup cobordism on each trivial circle, with some cups possibly carrying dots as specified by the labels $\mathcal {J}$. Diagrammatically, we will use the same convention as in Figure 3.

The following lemma concerns general (not necessarily standard) configurations; the argument is similar to the proof of [Reference Beliakova, Putyra and WehrliBPW19, Lemma 6.4].

Proof. Isotopic cobordisms are equal in $\mathcal {BN}_{\mathfrak {q}}(\mathbb {A})$ if the isotopy between them fixes the membrane. We may therefore assume that the isotopy $\phi$ is a sequence of the local moves in Figure 11, denoted $P, P^{-1}, N,$ and $N^{-1}$.

Figure 11. The moves $P, P^{-1}, N, N^{-1}$.

Let $p$ denote the number of moves of type $P$ or $P^{-1}$, let $n$ denote the number of moves of type $N$ or $N^{-1}$, and set $k=n-p$. It follows from the relations in Figure 7 that

\[ \mathfrak{q}^k\bar{S}S = \text{id}_{\mathcal{C}},\quad \mathfrak{q}^k S\bar{S} = \text{id}_{\mathcal{C}'}. \]

Proof. As discussed earlier, a standard generator $x= v_\mathcal {I} \otimes w_\mathcal {J} \in \mathcal {C}$ is the image of $v_\mathcal {I}$ under a cobordism

\[ \Sigma_\mathcal{J} : \mathcal{C}_{\rm E} \to \mathcal{C}, \]

which is the identity on $\mathcal {C}_\textrm {E}$ and a cup cobordism on all circles in $\mathcal {C}_\textrm {T}$, with some cups possibly carrying dots as specified by $\mathcal {J}$. Every component of the cobordism $S \Sigma _\mathcal {J}$ is either an undotted annulus between essential circles or a possibly dotted disk with trivial boundary. Each disk can be isotoped to a cup cobordism on its trivial boundary circle at the cost of multiplying by a power of $\mathfrak {q}$. Then, at the cost of introducing further powers of $\mathfrak {q}$, the remaining annuli may be isotoped to the identity cobordism on $\mathcal {C}_\textrm {E}$ while fixing each cup cobordism. This yields $\mathfrak {q}^k S\Sigma _\mathcal {J} = \Sigma _\mathcal {J}$ for some $k\in \mathbb {Z}$, so $\mathfrak {q}^k S(x) = \mathfrak {q}^k S \Sigma _\mathcal {J}(v_\mathcal {I}) = \Sigma _\mathcal {J}(v_\mathcal {I}) = x$.

Proof. Consider the component-preserving cobordism $\overline {S_1}S_2: \mathcal {C}^\circ \to \mathcal {C}^\circ$, which is formed by the isotopy $\phi _1^{-1} \phi$. By Lemma 2.7, we have $\mathfrak {q}^m \overline {S_1}S_2 (x) = x$ for some $m\in \mathbb {Z}$. By Lemma 2.6, we know that $\mathfrak {q}^\ell \overline {S_1} S_1 = \text {id}_{\mathcal {C}}$ for some $\ell \in \mathbb {Z}$. Then

\[ \mathfrak{q}^\ell \overline {S_1} S_1 (x) = x = \mathfrak{q}^m \overline {S_1}S_2(x) \]

and we obtain

\[ S_1(x) = \mathfrak{q}^{m-\ell} S_2(x). \]

So far the discussion concerned only generators of standard configurations. Next we consider generators for arbitrary configurations.

By Lemma 2.8, these generators of $\mathcal {C}$ are well defined up to multiplication by a (non-uniform, according to Remark 2.9) power of $\mathfrak {q}$, and also a possible re-ordering of the trivial circles in $\mathcal {C}^\circ$ which corresponds to a permutation of the indexing set $\mathcal {J}$. We assume throughout that there is a fixed isotopy $\mathcal {C}^\circ \to \mathcal {C}$, which will often not be named. Likewise, an unnamed cobordism $\mathcal {C}\to \mathcal {C}^\circ$ denotes the inverse of $\mathcal {C}^\circ \to \mathcal {C}$.

As discussed earlier, a standard generator $x\in \mathcal {C}^\circ$ is the image of the corresponding standard generator of $\mathcal {C}^\circ _\textrm {E}$ under a cobordism $\Sigma _\mathcal {J} : \mathcal {C}^\circ _\textrm {E} \to \mathcal {C}^\circ$, where $\Sigma _\mathcal {J}$ is the identity on $\mathcal {C}^\circ _\textrm {E}$ and a cup cobordism on each circle in $\mathcal {C}^\circ _\textrm {T}$, with some cups possibly carrying dots. Up to a power of $\mathfrak {q}$, the cobordism $S\Sigma _\mathcal {J}$ represents a cobordism which traces out an isotopy $\mathcal {C}^\circ _\textrm {E} \to \mathcal {C}_\textrm {E}$ and is a cup cobordism on each circle in $\mathcal {C}_\textrm {T}$. Then each standard generator of $\mathcal {C}$ can also be realized as the image of a cup cobordism on each trivial circle and an isotopy on the essential circles.

Note also that we do not make any assumptions about how these isotopies are picked for different configurations within a single cube of resolutions.

2.4 Computation of the saddle maps

In this subsection, we compute saddle maps in quantum annular homology using the relations in $\mathcal {BBN}_\mathfrak {q}(\mathbb {A})$ and Theorem 2.5. These results will be used in the formulation of the quantum annular Burnside functor in § 4.

We start with several examples; the general case is treated in Proposition 2.16. Saddle maps for various types of configurations (where intersections with the seam are minimal) are summarized in Figure 12. In the first two examples the calculation relies on the Boerner relation and relations satisfied by cobordisms in the Bar-Natan category. Specifically, one uses the neck-cutting relation and delooping; cf. [Reference Beliakova, Putyra and WehrliBPW19, Proposition 5.3]. (Note that delooping makes sense only for trivial, and not for essential, circles in the annulus.)

Figure 12. Surgery formulas in quantum annular Khovanov homology for curves with minimal intersections with the seam.

To analyze our saddle maps, we will use the language of surgery arcs, as in [Reference Lipshitz and SarkarLS14a, § 2]. For a configuration $\mathcal {C}$, a surgery arc is an interval embedded in $\mathbb {A}$ whose endpoints lie on $\mathcal {C}$ and whose interior is disjoint from $\mathcal {C}$. In the construction of quantum annular homology, link diagrams are assumed transverse to $\mu$ and all crossings are away from $\mu$. In light of this, we will assume that surgery arcs are disjoint from the seam. For a configuration $\mathcal {C}$ with a surgery arc, let $s(\mathcal {C})$ denote the configuration obtained by surgery on the arc. There is a saddle cobordism $\mathcal {C}\to s(\mathcal {C})$, which is well defined in $\mathcal {BN}_{\mathfrak {q}}(\mathbb {A})$. In terms of the cube of resolutions of a link diagram, a surgery arc may be placed at a $0$-smoothing to indicate that there will be a saddle cobordism at that smoothing.

Example 2.10 For the saddle

between standard configurations, we have the following formulas:

Algebraically, this is written as

\begin{align*} & v_+\otimes w_+ \mapsto v_+, \quad v_+\otimes w_- \mapsto 0, \\ & v_-\otimes w_+ \mapsto v_-, \quad v_-\otimes w_- \mapsto 0. \end{align*}

These can be deduced from Boerner's relation (Figure 5) and the fact that the two standard generators of a trivial circle are picked out by an undotted cup and a once-dotted cup.

Example 2.11 For the saddle

we have the formulas

which, algebraically, can be written as

\begin{align*} & V_+ \mapsto v_+\otimes w_-, \quad v_- \mapsto v_-\otimes w_-. \end{align*}

This can be deduced by cutting the neck along the trivial circle, which splits off as a result of the saddle, and then applying Boerner's relation.

The next two examples are also discussed in [Reference Beliakova, Putyra and WehrliBPW19, § 6.4].

Example 2.12 Let $S$ denote the following saddle:

Let $C$ denote the trivial circle on the right-hand side above. Since $C$ is not standard, we need to pick an isotopy to specify its generators. For the sake of calculation, we pick the following isotopy:

which specifies generators, represented diagrammatically, as

These generators are the images of $1\in \Bbbk$ under the undotted and dotted cup cobordisms on $C$, respectively. Since the cup cobordism on $C$ intersects the membrane, the placement of the dot is relevant, and the diagram shows where the dot is placed.

Now let $\Sigma$ denote the undotted cap cobordism on $C$, and let $\Sigma '$ denote the dotted cap cobordism on $C$, with the dot placed as in the generator $w_-$. Using the relations in $\mathcal {BN}_{\mathfrak {q}}(\mathbb {A})$, we obtain

\begin{align*} &\ \, \Sigma(w_+) = 0, \qquad \ \Sigma(w_-) = \mathfrak{q}^{{-}1}, \\ & \ \Sigma'(w_+) = \mathfrak{q}^{{-}1}, \quad \Sigma'(w_-) = 0.\hphantom{00} \end{align*}

Composing with the saddle $S$, observe that $\Sigma S = S^1\times \cap$ in $\mathcal {BBN}_\mathfrak {q}(\mathbb {A})$, and that $\Sigma 'S = 0$ by Boerner's relation. We are now in a position to write down formulas for $S$. For example, we may write

(2.7)\begin{equation} S(v_+\otimes v_-) = \alpha w_++ \beta w_- \end{equation}

for some $\alpha , \beta \in \Bbbk$. Applying $\Sigma$ to the above equality, we obtain

\[ \Sigma S(v_+\otimes v_-) = \mathfrak{q}^{{-}1} \beta. \]

Theorem 2.5 tells us that $\Sigma S(v_+ \otimes v_-) = ev(v_+ \otimes v_-) = \mathfrak {q}$, so that $\beta = \mathfrak {q}^2$. By applying $\Sigma '$ to both sides of 2.7, we obtain $\alpha = 0$. A similar argument for the remaining generators yields the full table of formulas for $S$:

Equivalently,

\begin{align*} S(v_+\otimes v_-) & = \mathfrak{q}^2 w_-, \quad S(v_+\otimes v_+) = 0, \\ S(v_-\otimes v_+) & = \mathfrak{q} w_-, \quad\ S(v_-\otimes v_-) = 0. \end{align*}

Example 2.13 Let $S$ denote the following saddle:

Pick generators $w_+$ and $w_-$ for the left-hand trivial circle $C$ as in Example 2.12. Let $\Sigma$ and $\Sigma '$ be the undotted and dotted cups on $C$, so that $w_+ = \Sigma (1)$ and $w_- = \Sigma '(1)$. Observe that $S \Sigma ' = 0$ by Boerner's relation, so

\[ S(w_-) = 0. \]

Finally, note that $S \Sigma = S^1 \times \cup$. By Theorem 2.5, we obtain

\[S(w_+) = v_+\otimes v_-+ \mathfrak{q}^{{-}1} v_-\otimes v_+.\]

Diagrammatically, the formulas for $S$ are

Saddle maps for various topological types of configurations, including the result of calculations in Examples 2.10–2.13, are summarized in Figure 12. The next example illustrates a calculation of the saddle map in a case of higher multiplicity of intersections between the configuration and the seam.

Example 2.14 Here is a slightly more involved version of Example 2.12. Consider the configurations $\mathcal {C}_1$ and $\mathcal {C}_2$, and the saddle $S: \mathcal {C}_1 \to \mathcal {C}_2$ as shown in Figure 13.

Figure 13. The saddle $S\colon {\mathcal C}_1\to {\mathcal C}_2$ in Example 2.14.

Fix generators for $\mathcal {C}_1$ and $\mathcal {C}_2$ using the isotopies $S_1$ and $S_2$ depicted in Figures 14 and 15. Since generators of $\mathcal {C}_2$ are the images of the standard generators of $\mathcal {C}_2^\circ$ under the isomorphism $S_2: \mathcal {C}_2^\circ \to \mathcal {C}_2$, it suffices to write down formulas for the composition

(2.8)\begin{equation} \mathcal{C}^\circ_1 \xrightarrow{S_1} \mathcal{C}_1 \xrightarrow{S} \mathcal{C}_2 \xrightarrow{S_2^{{-}1}} \mathcal{C}^\circ_2. \end{equation}

Note that $S_2^{-1} = \mathfrak {q}^3\overline{S_2}$ (see Lemma 2.6). Let $\Phi$ denote the composition (2.8).

Figure 14. An isotopy $S_1: \mathcal {C}_1^\circ \to \mathcal {C}_1$.

Figure 15. An isotopy $S_2: \mathcal {C}_2^\circ \to \mathcal {C}_2$.

Let $\Sigma$ and $\Sigma '$ be the undotted and dotted cap cobordisms, respectively, on the trivial circle $\mathcal {C}_2^\circ$. Note that $\Sigma '\Phi = 0$ by Boerner's relation. Applying a trace move from Figure 7 to the part of the cobordism depicted in (

2.9

), we see that

\[ \Sigma\overline{S_2}SS_1 \]

is equal to $\mathfrak {q}^{-1}(S^1\times \cap )$, so that

\[ \Sigma\Phi = \mathfrak{q}^2 (S^1\times{\cap}). \]

Arguing as in Example 2.12, we obtain

\begin{align*} & S(v_+\otimes v_-) = \mathfrak{q}^3w_-, \quad S(v_+\otimes v_+) = 0, \\ & S(v_-\otimes v_+) = \mathfrak{q}^2 w_-, \quad S(v_-\otimes v_-) = 0. \end{align*}

(2.9)

Remark 2.15 Here is a slightly different way to finish the computation in Example 2.14, which will be used in Proposition 2.16. In the morphism $\Phi : \mathcal {C}_1^\circ \to \mathcal {C}_2^\circ$, we may cut the neck along a small push-off of the trivial circle $\mathcal {C}_2^\circ$ to write $\Phi$ as a sum of two dotted cobordisms. One of the summands is $0$ by Boerner's relation, and the other is isotopic to a disjoint union of

\[ S^1 \times{\cap} \]

and a dotted cup cobordism on $\mathcal {C}_2^\circ$, which is denoted $\overline {\Sigma '}$ using the notation of Example 2.14. Then, using the trace relations in $\mathcal {BN}_{\mathfrak {q}}(\mathbb {A})$, we see that

\[ \Phi = \mathfrak{q}^2 (S^1 \times{\cap}) \sqcup \overline{\Sigma '} \]

and the formulas in Example 2.14 follow.

Example 2.14 shows that there is considerable complexity in computing the saddle map when curves have multiple intersections with the seam. The next proposition extends Examples 2.12–2.14 to the case of arbitrary configurations. It will be important for the analysis in § 2.5. Recall the numbering of circles discussed in the paragraph preceding (2.6).

Proposition 2.16 Let $\mathcal {C}$ be a configuration with a surgery arc $A$. Let $S: \mathcal {C} \to s(\mathcal {C})$ denote the saddle.

(1) Suppose that both endpoints of $A$ are on a trivial circle $C$, and that surgery along $A$ splits $C$ into two essential circles. Assume that $C$ is first in the ordering on trivial circles of $\mathcal {C}$, and it splits into the $i$th and $(i+1)$th essential circles in $s(\mathcal {C})$. Let $x= v_{\mathcal {I}'} \otimes v_{\mathcal {I}''} \otimes w_+ \otimes w_\mathcal {J}$ be a generator of $\mathcal {C}$ in which $C$ is undotted, where $\mathcal {I}'$ labels the first $i-1$ essential circles. Then

\[ S(x) = \mathfrak{q}^a v_{\mathcal{I}'} \otimes v_+\otimes v_-\otimes v_{\mathcal{I}''}\otimes w_\mathcal{J}+ \mathfrak{q}^{a-1} v_{\mathcal{I}'} \otimes v_-\otimes v_+\otimes v_{\mathcal{I}''}\otimes w_\mathcal{J} \]

for some $a\in \mathbb {Z}$.

(2) Suppose that the endpoints of $A$ are on the $i$th and $(i+1)$th essential circles of $\mathcal {C}$. Consider the generators

\begin{align*} y_1 &= v_{\mathcal{I}'} \otimes v_+\otimes v_-\otimes v_{\mathcal{I}''} \otimes w_\mathcal{J},\\ y_2 &= v_{\mathcal{I}'} \otimes v_-\otimes v_+\otimes v_{\mathcal{I}''} \otimes w_\mathcal{J} \end{align*}

of $\mathcal {C}$, where $\mathcal {I}'$ labels the first $i-1$ essential circles. Then

\begin{align*} S(y_1) &= \mathfrak{q}^{b+1} v_{\mathcal{I}'} \otimes v_{\mathcal{I}''} \otimes w_-\otimes w_\mathcal{J}, \\ S(y_2) &= \mathfrak{q}^b v_{\mathcal{I}'} \otimes v_{\mathcal{I}''} \otimes w_-\otimes w_\mathcal{J} \end{align*}

for some $b\in \mathbb {Z}$.

Proof. For both $(1)$ and $(2)$, it is enough to show that the result holds after applying $s(\mathcal {C}) \to s(\mathcal {C})^\circ$.

(1). The generator $x$ is the image of $v_{\mathcal {I}'}\otimes v_{\mathcal {I}''}$ under the composition $\mathcal {C}_\textrm {E}^\circ \xrightarrow {\Sigma _\mathcal {J}} \mathcal {C}^\circ \to \mathcal {C}$, where $\Sigma _\mathcal {J}$ is a cup cobordism on trivial circles in $\mathcal {C}^\circ$, with the cups dotted according to $\mathcal {J}$. Note that the cup on $C$ is undotted. Let $\Psi$ denote the composition

\[ \mathcal{C}^\circ_{\rm E} \xrightarrow{\Sigma_\mathcal{J}} \mathcal{C}^\circ \to \mathcal{C} \xrightarrow{S} s(\mathcal{C}) \to s(\mathcal{C})^\circ. \]

The cobordism $\Psi$ is isotopic to a disjoint union of

and cup cobordisms on each trivial circle in $s(\mathcal {C})^\circ$, with dots placed according to $\mathcal {J}$. By Theorem 2.5, the cobordism $(*)$ induces the map

\[ v_{\mathcal{I}'} \otimes v_{\mathcal{I}''} \mapsto v_{\mathcal{I}'}\otimes v_+\otimes v_-\otimes v_{\mathcal{I}''} + \mathfrak{q}^{{-}1} v_{\mathcal{I}'} \otimes v_-\otimes v_+\otimes v_{\mathcal{I}''} \]

and the result follows.

(2). The generators $y_1$ and $y_2$ are the images of $v_{\mathcal {I}'} \otimes v_+ \otimes v_-\otimes v_{\mathcal {I}''}$ and $v_{\mathcal {I}'} \otimes v_- \otimes v_+ \otimes v_{\mathcal {I}''}$, respectively, under

\[ \mathcal{C}^\circ_{\rm E} \xrightarrow{\Sigma_\mathcal{J}} \mathcal{C}^\circ \to \mathcal{C}, \]

where $\Sigma _\mathcal {J}$ is a cup cobordism on trivial circles with dots placed according to $\mathcal {J}$. Let $\Phi$ denote the composition

\[ \mathcal{C}^\circ_{\rm E} \xrightarrow{\Sigma_\mathcal{J}} \mathcal{C}^\circ \to \mathcal{C} \xrightarrow{S} s(\mathcal{C}) \to s(\mathcal{C})^\circ. \]

Let $C\in s(\mathcal {C})$ denote the trivial circle obtained by surgery along $A$ and let $C'\in s(\mathcal {C})^\circ$ denote the corresponding circle. In the cobordism $\Phi$, we may cut the neck along $C'$ to write $\Phi$ as a sum of two cobordisms. One of the summands is $0$ by Boerner's relation, and the other is isotopic to the disjoint union of

and cup cobordisms on each trivial circle. Observe also that the cup cobordism on $C'$ is dotted. Finally, Theorem 2.5 says that the cobordism $(**)$ induces the map

\begin{align*} v_{\mathcal{I}'} \otimes v_+\otimes v_-\otimes v_{\mathcal{I}''} & \mapsto \mathfrak{q} v_{\mathcal{I}'} \otimes v_{\mathcal{I}''},\\ v_{\mathcal{I}'} \otimes v_-\otimes v_+\otimes v_{\mathcal{I}''} & \mapsto v_{\mathcal{I}'} \otimes v_{\mathcal{I}''} \end{align*}

and the result follows.

We end this subsection with a discussion about recovering classical annular homology. Consider the map $\Bbbk \to \mathbb {Z}$ which is the identity on $\mathbb {Z} \subset \Bbbk$ and sends $\mathfrak {q}$ to $1$. It induces a functor $(-) \otimes _{\Bbbk } \mathbb {Z} : \text {Mod}(\Bbbk ) \to \text {Mod}(\mathbb {Z})$. Thus, one can consider the composition

\[ \mathcal{BBN}_{\mathfrak{q}}(\mathbb{A}) \xrightarrow{\mathcal{F}_{\mathbb{A}_{\mathfrak{q}}}} \text{Mod}(\Bbbk) \to \text{Mod}(\mathbb{Z}), \]

which we denote $\mathcal {F}_{\mathbb {A}_{\mathfrak {q}}} \otimes _{\Bbbk } \mathbb {Z}$. Tensoring with $\mathbb {Z}$ forgets the action of $\mathfrak {q}$, in the sense that isotopic cobordisms induce equal maps under $\mathcal {F}_{\mathbb {A}_{\mathfrak {q}}} \otimes _{\Bbbk } \mathbb {Z}$ even when the isotopy is not required to fix the seam. Let $\mathcal {C}$ be a configuration with $e$ essential and $t$ trivial circles. By Lemma 2.8, there is a canonical isomorphism

\[ \mathcal{F}_{\mathbb{A}_{\mathfrak{q}}}(\mathcal{C}) \otimes_{\Bbbk} \mathbb{Z} \cong V_\mathbb{Z}^{\otimes e} \otimes_\mathbb{Z} W_\mathbb{Z}^{\otimes t} \]

obtained by picking an isotopy $\mathcal {C}^\circ \to \mathcal {C}$. It is implicit in [Reference Beliakova, Putyra and WehrliBPW19] that $\mathcal {F}_{\mathbb {A}_{\mathfrak {q}}} \otimes _{\Bbbk } \mathbb {Z}$ is the classical annular functor $\mathcal {F}_\mathbb {A}$; indeed, this is straightforward to check using the relations in $\mathcal {BBN}_\mathfrak {q}(\mathbb {A})$ and Theorem 2.5 as in the proof of Proposition 2.16.

Lemma 2.17 Let $\mathcal {C}$ be a configuration with a single surgery arc and let $S: \mathcal {C} \to s(\mathcal {C})$ denote the saddle. Let $x\in \mathcal {C}$ be a generator. Then

\[ S(x) = \sum \varepsilon_y y, \]

where the sum is over generators of $s(\mathcal {C})$ and each $\varepsilon _y$ is either $0$ or a power of $\mathfrak {q}$. Moreover, $\varepsilon _y \neq 0$ if and only if $y$ appears in $\mathcal {F}_{\mathbb {A}}(S)(x)$, where $\mathcal {F}_\mathbb {A}$ is the classical (unquantized) annular TQFT.

2.5 Ladybug configurations

In this subsection, we analyze the ladybug configuration (see [Reference Lipshitz and SarkarLS14a, Definition 5.6]; see also [Reference Lipshitz and SarkarLS14a, Figure 5.1]). Examining ladybug configurations is crucial in the construction of Khovanov homotopy types in various contexts. We start with a discussion of ladybug configurations in classical annular homology. We will then examine a particular type of ladybug configuration in quantum annular homology, and we will indicate how the analysis differs from that in classical annular homology.

We recall the notion of a ladybug configuration. A circle $C\subset \mathbb {A}$ with two surgery arcs forms a ladybug configuration if the endpoints of the two arcs alternate around $C$. We will say that a configuration $\mathcal {C}$ with surgery arcs has a ladybug configuration if a circle $C$ in $\mathcal {C}$ and two of the surgery arcs form a ladybug configuration.

First, consider ladybug configurations in classical annular homology. Let $C\subset \mathbb {A}$ be a circle carrying two surgery arcs $A_1$ and $A_2$ which form a ladybug configuration. Figure 16 illustrates the three possibilities in the annulus.

Figure 16. Three ladybug configurations in the annulus.

For $i=1,2$, let $\mathcal {C}_i$ denote the configuration obtained by performing surgery along $A_i$, and let $d_i : \mathcal {F}_\mathbb {A}(C) \to \mathcal {F}_\mathbb {A}(\mathcal {C}_i)$ denote the maps assigned to the saddles in classical annular Khovanov homology. Let $\mathcal {C}'$ denote the final configuration, obtained by performing surgery on $C$ along both $A_1$ and $A_2$. When $C$ is essential, as in Figure 16(a), composing two saddle maps yields $0$ (see the formulas in Figure 4). Now consider the cases where $C$ is trivial, as in Figure 16(b) and 16(c). The dotted generator $w_-$ is sent to $0$ by the composition of two saddle maps. On the other hand, the two summands appearing in each of $d_1(w_+)$ and $d_2(w_+)$ are mapped to the same element in the final configuration $\mathcal {C}'$. The case of Figure 16(c) is illustrated in Figure 17.

Figure 17. The analysis in the case of Figure 16(c).

When constructing stable homotopy refinements in this framework, it is important to have a bijection between these intermediate generators which appear in $d_1(w_+)$ and $d_2(w_+)$, such that these bijections are coherent, in an appropriate sense, within higher dimensional cubes (cf. § 3.2 below). The bijections are the ladybug matchings defined in [Reference Lipshitz and SarkarLS14a, § 5.4].

In the case of Figure 16(b), the annulus and seam play no role, and the ladybug matching from [Reference Lipshitz and SarkarLS14a] can be used without significant alteration for both classical and quantum annular homology. The remainder of this subsection examines the ladybug configuration of the type in Figure 16(c) in quantum annular homology.

Let $\mathcal {C}$ be a configuration with two surgery arcs $A_\textrm {T}$ and $A_\textrm {E}$, both having endpoints on a trivial circle $C$ in $\mathcal {C}$. Suppose also that surgery along $A_\textrm {T}$ splits $C$ into two trivial circles and surgery along $A_\textrm {E}$ splits $C$ into two essential circles; cf. Figure 18. Let $s_\textrm {T}(\mathcal {C})$ and $s_\textrm {E}(\mathcal {C})$ denote the configurations obtained by surgery along $A_\textrm {T}$ and $A_\textrm {E}$, respectively. Let $\mathcal {C}'$ denote the final configuration, obtained by surgery along both arcs. Let $C'\in \mathcal {C}'$ denote the circle obtained by both surgeries. We have the following commutative square.

(2.10)

Figure 18 exhibits the specific instance of this set-up corresponding directly to Figure 16(c), but there are many such cases depending on how $C$ intersects the seam. As usual, we will not distinguish between cobordisms and their induced maps. We assume that $C$ occurs first in the ordering on trivial circles in $\mathcal {C}$. Surgery along $A_\textrm {T}$ splits $C$ into two trivial circles in $s_\textrm {T}(\mathcal {C})$, which we assume are the first two trivial circles in $s_\textrm {T}(\mathcal {C})$. Finally, we order these first two circles as follows. Orient the arc $A_\textrm {T}$ such that it points from the outer essential circle in $s_\textrm {E}(\mathcal {C})$ to the inner one. Declare that the first circle $C_1$ is to the left of $A_\textrm {T}$ and the second $C_2$ is to the right of $A_\textrm {T}$. Our ordering convention is illustrated in Figure 19.

Figure 18. Further analysis in the case of Figure 16(c).

Figure 19. Ordering convention on $C_1$ and $C_2$.

Let $x= v_\mathcal {I} \otimes w_+ \otimes w_\mathcal {J} \in \mathcal {C}$ be a generator in which $C$ is undotted. By Lemma 2.17, we obtain

\[ \Delta_{\rm T}(x) = \mathfrak{q}^k v_\mathcal{I} \otimes w_-\otimes w_+\otimes w_\mathcal{J} + \mathfrak{q}^\ell v_\mathcal{I} \otimes w_+\otimes w_-\otimes w_\mathcal{J} \]

for some $k,\ell \in \mathbb {Z}$.

The following corollary implies that, in fact, in the quantum annular setting, there is no need for a ladybug matching for this ladybug configuration because intermediate generators are mapped to different elements in the final configuration.

Corollary 2.18 With the above notation,

\[ m_{\rm E}(\Delta_{\rm E}(x)) = \mathfrak{q}^m v_\mathcal{I} \otimes w_-\otimes w_\mathcal{J} + \mathfrak{q}^{m+2} v_\mathcal{I} \otimes w_-\otimes w_\mathcal{J} \]

for some $m\in \mathbb {Z}$. Moreover, one of $m_{\rm T}(\mathfrak {q}^k v_\mathcal {I} \otimes w_-\otimes w_+ \otimes w_\mathcal {J})$ or $m_{\rm T}(\mathfrak {q}^\ell v_\mathcal {I} \otimes w_+ \otimes w_- \otimes w_\mathcal {J})$ is equal to $\mathfrak {q}^m v_\mathcal {I} \otimes w_-\otimes w_\mathcal {J}$ and the other is equal to $\mathfrak {q}^{m+2} v_\mathcal {I} \otimes w_-\otimes w_\mathcal {J}$.

It will be important for § 7 to know which of $m_\textrm {T}(\mathfrak {q}^k v_\mathcal {I} \otimes w_-\otimes w_+ \otimes w_\mathcal {J})$ or $m_\textrm {T}(\mathfrak {q}^\ell v_\mathcal {I} \otimes w_+ \otimes w_- \otimes w_\mathcal {J})$ is equal to $\mathfrak {q}^m v_\mathcal {I} \otimes w_-\otimes w_\mathcal {J}$. This is addressed in the following proposition.

Proposition 2.20 With the above notation,

\begin{align*} m_{\rm T}(\mathfrak{q}^k v_\mathcal{I} \otimes w_-\otimes w_+\otimes w_\mathcal{J}) &=\mathfrak{q}^m v_\mathcal{I} \otimes w_-\otimes w_\mathcal{J},\\ m_{\rm T}(\mathfrak{q}^\ell v_\mathcal{I} \otimes w_+\otimes w_-\otimes w_\mathcal{J}) &= \mathfrak{q}^{m+2} v_\mathcal{I} \otimes w_-\otimes w_\mathcal{J}. \end{align*}

Proof. The generator $x = v_\mathcal {I} \otimes w_+ \otimes w_\mathcal {J}$ is the image of $v_\mathcal {I}$ under $\mathcal {C}^\circ _\textrm {E} \xrightarrow {\Sigma _\mathcal {J}} \mathcal {C}^\circ \xrightarrow {S} \mathcal {C}$. Here $\Sigma _\mathcal {J}$ is the usual identity cobordism on the essential part together with an undotted cup corresponding to $w_+$ on $C$ and various other cups dotted according to $\mathcal {J}$, while $S$ is some chosen cobordism from the standard $\mathcal {C}^\circ$ to $\mathcal {C}$. Isotope the resulting disk in $S\Sigma _\mathcal {J}$ bounding $C$ to a cup cobordism on $C$ to obtain a new cobordism $\Sigma ' : \mathcal {C}^\circ _\textrm {E} \to \mathcal {C}$, so that $S\Sigma _\mathcal {J} =\mathfrak {q}^c \Sigma '$ for some $c\in \mathbb {Z}$.

Since $C$ is trivial, it bounds a disk $D$ in the annulus. Note that $A_\textrm {T}$ lies inside $D$, so we may push it into the cup cobordism on $C$ in $\Sigma '$ to obtain an arc $A$. We may also pull $A_\textrm {T}$ onto the saddle $\Delta _\textrm {T}$ to obtain another arc $A'$. Performing neck cutting on the circle $A\cup A'$ on $S\Sigma '$ yields

\[ \Delta_{\rm T} \Sigma' = S_1 + S_2, \]

where $S_1$ and $S_2$ are labelled such that $S_1$ is dotted on $C_1$ and $S_2$ is dotted on $C_2$. Figure 20 illustrates the local picture near $A_\textrm {T}$; the surgery arc $A_\textrm {T}$ is decorated by an arrowhead. From this we obtain

\[ \Delta_{\rm T}(x) = \Delta_{\rm T}S\Sigma_\mathcal{J}(v_\mathcal{I}) = \mathfrak{q}^c \Delta_{\rm T}\Sigma'(v_\mathcal{I}) = \mathfrak{q}^c S_1(v_\mathcal{I}) + \mathfrak{q}^c S_2(v_\mathcal{I}) \]

and it follows that

\begin{align*} \mathfrak{q}^cS_1(v_\mathcal{I}) &= \mathfrak{q}^k v_\mathcal{I} \otimes w_-\otimes w_+\otimes w_\mathcal{J},\\ \mathfrak{q}^cS_2(v_\mathcal{I}) &= \mathfrak{q}^\ell v_\mathcal{I} \otimes w_+\otimes w_-\otimes w_\mathcal{J}. \end{align*}

Figure 20. Neck cutting in the proof of Proposition 2.20.

The relation

implies that $\mathfrak {q}^d m_\textrm {T}S_1 = m_\textrm {T}S_2$ for some $d\in \mathbb {Z}$. To compute $\mathfrak {q}^d$, we need to move the dot on $S_2$ along $C'$ until it is in the same position as the dot on $S_1$, and count (with sign) the number of times the dot intersects the membrane during this process. This is the same as the signed intersection between the seam and one of the essential circles in $s_\textrm {E}(\mathcal {C})$ obtained by surgery on $C$. The situation is depicted below in (

2.11

); we need to move the dot on the right-hand diagram along the circle, without intersecting the surgery arc, to the other side of the arc.

(2.11)

Our convention of ordering $C_1$ and $C_2$ (see Figure 19) guarantees that the dot is moved counterclockwise along an essential circle in $s_\textrm {E}(\mathcal {C})$, so that $m_\textrm {T}S_2= \mathfrak {q}^2 m_\textrm {T}S_1$. Finally, this implies that

\[ \mathfrak{q}^2 m_{\rm T}(\mathfrak{q}^k v_\mathcal{I} \otimes w_-\otimes w_+\otimes w_\mathcal{J}) = m_{\rm T}(\mathfrak{q}^\ell v_\mathcal{I} \otimes w_+\otimes w_-\otimes w_\mathcal{J}) \]

and the result follows.

There is always either a ‘right’ or a ‘left’ choice which is made at the very beginning of defining the ladybug matching (see [Reference Lipshitz and SarkarLS14a, § 5.4]). Consider the classical annular differential for the ladybug configuration of Figure 17. The ladybug matching made with the left choice identifies the circles in the middle smoothings as shown in Figure 21(a). Then the ladybug matching pairs up the intermediate generators appearing in $\Delta _\textrm {T}(w_+)$ and $\Delta _\textrm {E}(w_+)$ as shown in Figure 21(b).

Figure 21. The left choice in the ladybug matching.

Now consider the quantum annular surgery formulas for the same configuration (Figure 18), which are detailed in Figure 22. We see that the intermediate generators get paired up as in (

):

(2.12)

Algebraically, the matching is

\begin{align*} &w_+\otimes w_- \longleftrightarrow v_+\otimes v_-,\\ &w_-\otimes w_+ \longleftrightarrow \mathfrak{q}^{{-}1} v_-\otimes v_+, \end{align*}

where the ordering on trivial circles follows the convention illustrated in Figure 19. After forgetting powers of $\mathfrak {q}$, the matching in (

) is consistent with the matching in Figure 21. Our remaining goal in this section is to show that the matching forced by powers of $\mathfrak {q}$ agrees with the ladybug matching made with the left pair, as stated in Proposition 2.21.

Figure 22. Note that the term with coefficient ${\mathfrak {q}}^2$ on bottom right matches the term directly above it, because dragging the dot across the seam amounts to multiplication by ${\mathfrak {q}}^2$.

We will use the notation and conventions established in this section. By Proposition 2.16(1), we can write

\[ \Delta_{\rm E}(x) = \mathfrak{q}^a v_{\mathcal{I}'} \otimes v_+\otimes v_-\otimes v_{\mathcal{I}''}\otimes w_\mathcal{J}+ \mathfrak{q}^{a-1} v_{\mathcal{I}'} \otimes v_-\otimes v_+\otimes v_{\mathcal{I}''}\otimes w_\mathcal{J} \]

for some $a\in \mathbb {Z}$. Proposition 2.16(2), Corollary 2.18, and Proposition 2.20 imply that in the quantum setting, the pairing on intermediate generators is forced to be

(2.13)\begin{equation} \begin{gathered} \mathfrak{q}^\ell v_\mathcal{I} \otimes w_+\otimes w_-\otimes w_\mathcal{J} \longleftrightarrow \mathfrak{q}^a v_{\mathcal{I}'} \otimes v_+\otimes v_-\otimes v_{\mathcal{I}''}\otimes w_\mathcal{J}, \\ \mathfrak{q}^k v_\mathcal{I} \otimes w_-\otimes w_+\otimes w_\mathcal{J} \longleftrightarrow \mathfrak{q}^{a-1} v_{\mathcal{I}'} \otimes v_-\otimes v_+\otimes v_{\mathcal{I}''}\otimes w_\mathcal{J}. \end{gathered} \end{equation}

Proof. Looking at the surgery arc $A_\textrm {T}$, the left choice makes the following identification on circles in $s_\textrm {E}(\mathcal {C})$ and $s_\textrm {T}(\mathcal {C})$:

(2.14)

Therefore, the ladybug matching makes the following identification on generators:

(2.15)

Comparing with our ordering convention on the circles in $s_\textrm {T}(\mathcal {C})$ in Figure 19 demonstrates that this is consistent with (2.13).

3.1 The cube category

We first recall the cube category $\underline {2}^n$ from [Reference Lawson, Lipshitz and SarkarLLS20, § 2.1]. The objects of $\underline {2}^n$ are the elements of $\{0,1\}^n$, thought of as vertices of the $n$-dimensional cube $I^n$. There is a natural partial order on $\{0,1\}^n$: for vertices $u=(u_1,\ldots , u_n)$ and $v=(v_1, \ldots , v_n)$ in $\{0,1\}^n$, write $u\geq v$ if each $u_i \geq v_i$. The set of morphisms $\text {Hom}_{\underline {2}^n}(u,v)$ is defined to be empty unless $u\geq v$, in which case $\text {Hom}_{\underline {2}^n}(u,v)$ consists of a single element, denoted $\varphi _{u,v}$. Therefore, for $u\geq w$, we have $\varphi _{u,w} = \varphi _{v,w} \circ \varphi _{u,v}$ for any $v$ such that $u\geq v \geq w$. We note that the edges in $\underline {2}^n$ point in the opposite direction of those in the cube of resolutions of a link diagram.

For a vertex $u=(u_1,\ldots , u_n)$, define $\vert {u} \vert := \sum _i u_i$. Write $u\geq _k v$ if $u \geq v$ and $\vert {u} \vert - \vert {v} \vert = k$. In particular, $u \geq _1 v$ means there is an edge from $u$ to $v$ in the cube $I^n$. Vertices $u,v$ with $u\geq _k v$ specify a $k$-dimensional subcube, which is the full subcategory of $\underline {2}^n$ with objects consisting of all vertices $w$ satisfying $u\geq w \geq v$.

3.2 Burnside categories and functors

This section discusses the Burnside category, $\mathscr {B}$, and Burnside functors, following [Reference Lawson, Lipshitz and SarkarLLS20, § 4.1].

For sets $X$ and $Y$, a correspondence from $X$ to $Y$ is a triple $(A,s,t)$, where $A$ is a set and $s:A\to X$, $t: A\to Y$ are functions, called the source map and target map, respectively. We will often denote a correspondence by $X \xleftarrow {s} A \xrightarrow {t} Y$ or simply $X \leftarrow A \to Y$. Given correspondences $X \xleftarrow {s_A} A \xrightarrow {t_A} Y$ and $Y \xleftarrow {s_B} B \xrightarrow {t_B} Z$, their composition $(B, s_B, t_B) \circ (A, s_A, t_A)$ is the correspondence $(C, s, t)$ from $X$ to $Z$ obtained as the fiber product

\[ C = B\times_Y A = \{ (b,a) \in B\times A \mid s_B(b) = t_A(a) \}, \]

with the source and target maps

\[ s(b,a) = s_A(a), \quad t(b,a) = t_B(b). \]

The composition can be summarized by the fiber product diagram

A morphism from a correspondence $X \xleftarrow {s_A} A \xrightarrow {t_A} Y$ to a correspondence $X \xleftarrow {s_B} B \xrightarrow {t_B} Y$ is a bijection $f:A\to B$ which commutes with the source and target maps. That is, $f$ is a bijection fitting into the following commutative diagram.

The collection of sets, correspondences between them, and morphisms of correspondences forms a bicategory in the language of [Reference Beliakova, Putyra and WehrliBPW19] or, equivalently, a weak $2$-category in the language of [Reference Lawson, Lipshitz and SarkarLLS20]. The objects are sets, $1$-morphisms are correspondences, and $2$-morphisms are morphisms of correspondences. We will use the terms bicategory and weak $2$-category interchangeably. A quick reference for the notion of bicategories is [Reference Beliakova, Putyra and WehrliBPW19, Appendix A.4].

The identity $1$-morphism of a set $X$ is the identity correspondence

\[ X \xleftarrow{\text{id}_X} X \xrightarrow{\text{id}_X} X. \]

Given correspondences $X \xleftarrow {s_A} A \xrightarrow {t_A} Y$ and $Z \xleftarrow {s_B} B \xrightarrow {t_B} X$, the compositions $(A, s_A, t_A) \circ (X, \text {id}_X, \text {id}_X)$ and $(X, \text {id}_X, \text {id}_X) \circ (B, s_B, t_B)$ are not equal to $(A, s_A, t_A)$ and $(B, s_B, t_B)$, but there are natural $2$-morphisms

\begin{align*} &(A, s_A, t_A) \circ (X, \text{id}_X, \text{id}_X) \cong (A, s_A, t_A), \\ &(X, \text{id}_X, \text{id}_X) \circ (B, s_B, t_B) \cong (B, s_B, t_B). \end{align*}

Similarly, composition of correspondences is not strictly associative, but is associative up to natural isomorphism. This is the sense in which we have only a weak 2-category, as opposed to a strict 2-category.

The Burnside category, denoted $\mathscr {B}$, is the sub-bicategory of the above consisting of finite sets and finite correspondences. Even though $\mathscr {B}$ is a bicategory, it will always be referred to as a category.

The construction of Khovanov homotopy types in [Reference Lawson, Lipshitz and SarkarLLS20] and [Reference Sarkar, Scaduto and StoffregenSSS20] utilizes functors $F: \underline {2}^n \to \mathscr {B}$, which we explain here. First, make $\underline {2}^n$ into a (strict) $2$-category by introducing only identity $2$-morphisms. There is a notion of a lax 2-functor between $2$-categories, and also of a strictly unitary lax 2-functor. The complete definitions, consisting of a slew of data and various natural morphisms, can be found in [Reference Lawson, Lipshitz and SarkarLLS20, Definitions 4.2 and 4.3]. We will only be interested in the notion of a Burnside functor, which is a strictly unitary lax 2-functor $F: \underline {2}^n \to \mathscr {B}$. Lemma 3.1 specifies the data needed to define a Burnside functor uniquely up to natural isomorphism (see § 3.5 for the definition of natural isomorphisms).

Lemma 3.1 ([Reference Lawson, Lipshitz and SarkarLLS20, Lemma 4.5], [Reference Lawson, Lipshitz and SarkarLLS17a, Proposition 4.3])

Consider the following data.

• • A finite set $F(u)$ for each vertex $u\in \underline {2}^n$.

• • A finite correspondence $F(\varphi _{u,v})$ from $F(u)$ to $F(v)$ for each pair of vertices $u,v\in \underline {2}^n$ with $u\geq _1 v$.

• • A $2$-morphism

  \[ F_{u,v,v',w} : F(\varphi_{v,w}) \circ F(\varphi_{u,v}) \to F(\varphi_{v',w}) \circ F(\varphi_{u,v'}) \]
  for each two-dimensional face of $\underline {2}^n$ with vertices $u,v,v',w$ satisfying $u\geq _1 v, v' \geq _1 w$.

Suppose also that the above data satisfies the following conditions.

1. (1) $F_{u,v,v',w}^{-1} = F_{u,v',v,w}$.

2. (2) For every three-dimensional subcube of $\underline {2}^n$ as in Figure 23(a), the hexagon of Figure 23(b) commutes.

Then the data can be extended to a strictly unitary lax $2$-functor $F:\underline {2}^n \to \mathscr {B}$, which is unique up to natural isomorphism.

Figure 23. (a) Three-dimensional cube and (b) the hexagon relation.

The hexagon of Figure 23(b) comes from two ways of traversing the faces of the three-dimensional cube, starting from the correspondence $F(\varphi _{w,z}) \circ F(\varphi _{v,w}) \circ F(\varphi _{u,v})$ and ending at $F(\varphi _{w',z}) \circ F(\varphi _{v'',w'})\circ F(\varphi _{u,v''})$. The top half of the hexagon comes from traversing the faces as in Figure 24(a) and the bottom half comes from traversing the faces as in Figure 24(b). Lemma 3.1 states that the hexagon relation is enough to guarantee that the functor is coherent on all higher dimensional cubes as well.

Remark 3.2 We end this subsection with some comments about verifying the hexagon relation which will be useful in proving Theorem 4.2. Suppose that we have $2$-morphisms $F_{u,v,v',w}$ for each square face $u\geq _1 v,v' \geq _1 w$ of $\underline {2}^n$, which satisfy $F_{u,v,v',w}^{-1} = F_{u,v',v,w}$ as in condition (1) of Lemma 3.1. Then verifying the hexagon relation is equivalent to the following. Start at the correspondence $F(\varphi _{w,z}) \circ F(\varphi _{v,w}) \circ F(\varphi _{u,v})$ and traverse the six faces of the cube using the $2$-morphisms; i.e. first move across the three faces as shown in Figure 24(a) and then move across the remaining three faces as in Figure 24(b), except in the reverse order. Composing these six $2$-morphisms yields a $2$-morphism

\[ \Phi_{u,v,w,z}: F(\varphi_{w,z}) \circ F(\varphi_{v,w}) \circ F(\varphi_{u,v}) \to F(\varphi_{w,z}) \circ F(\varphi_{v,w}) \circ F(\varphi_{u,v}). \]

Verifying commutativity of the hexagon of Lemma 3.1 is equivalent to verifying that $\Phi _{u,v,w,z}$ is the identity. Moreover, for each three-dimensional subcube of $\underline {2}^n$, it suffices to verify that $\Phi _{u,v,w,z}=\text {id}$ for just one tuple of vertices $u\geq _1 v \geq _1 w \geq _1 z$ within the subcube.

Figure 24. Illustration of the hexagon relation in Remark 3.2.

Furthermore, such verifications are immediate under certain circumstances, which we now describe. Let $A$ denote the correspondence $F(\varphi _{w,z}) \circ F(\varphi _{v,w}) \circ F(\varphi _{u,v})$, and let $s: A\to F(u)$, $t: A\to F(z)$ denote the source and target maps, respectively. Suppose that for every $x\in F(u)$ and $y\in F(z)$, $s^{-1}(x) \cap t^{-1}(y)$ is either empty or has one element. Let $a\in A$ and let $a' = \Phi _{u,v,w,z}(a)$. Since $\Phi _{u,v,w,z}(a)$ is a $2$-morphism, we have $s(a') = s(a)$ and $t(a') = t(a)$. Then $a'=a$, so the hexagon relation is satisfied for this three-dimensional subcube. In this situation, we will say that this three-dimensional cube is simple.

3.3 The equivariant Burnside category

Let $G$ be a finite group. The $G$-equivariant Burnside category, denoted $\mathscr {B}_G$, is an equivariant analogue of $\mathscr {B}$. Objects of $\mathscr {B}_G$ are finite free $G$-sets. A $1$-morphism from $X$ to $Y$ is a triple $(A, s, t)$, where $A$ is another finite free $G$-set, and $s : A\to X$, $t : A\to Y$ are $G$-equivariant maps. We will call such a triple $(A,s,t$) an equivariant correspondence. Given equivariant correspondences $X\xleftarrow {s_A} A \xrightarrow {t_A} Y$ and $Y \xleftarrow {s_B} B \xrightarrow {t_B} Z$, the composition $(B, s_B, t_B) \circ (A, s_A, t_A)$ is the same as in $\mathscr {B}$. The $G$-action on $B\times _Y A$ is inherited from the diagonal $G$-action on $B\times A$; that is, $g(b,a) = (gb,ga)$. The additional requirement on $2$-morphisms between correspondences is that they be $G$-equivariant.

The equivariant Burnside category is discussed in [Reference Sarkar, Scaduto and StoffregenSSS20, § 3.3] in the case $G= \mathbb {Z}_2$. Lemma 3.2 in [Reference Sarkar, Scaduto and StoffregenSSS20] (our Lemma 3.1) gives sufficient conditions for defining a functor $F:\underline {2}^n \to \mathscr {B}_{\mathbb {Z}_2}$, and the same conditions clearly work for general $G$. The modification to the data of Lemma 3.1 is that all sets should be finite free $G$-sets and all set maps should be equivariant. Note that if $G=\{1\}$, then $\mathscr {B}_{\{1\}} = \mathscr {B}$, so everything stated about $\mathscr {B}_G$ in the following sections holds just as well for $\mathscr {B}$.

We will later be interested in the quotient functor $(-)/G : \mathscr {B}_G\to \mathscr {B}$, which simply takes the quotient of all sets and set maps. Explicitly, the quotient functor sends a $G$-set $X$ to the set of orbits $X/G = X/(x\sim gx)$. For $G$-sets $X$ and $Y$ and an equivariant map $f:X\to Y$, there is an induced map $f/G : X/G \to Y/G$, given by $(f/G)([x]) = [f(x)]$. The quotient functor sends an equivariant correspondence $X\xleftarrow {s} A \xrightarrow {t} Y$ to the correspondence $X/G \xleftarrow {s/G} A/G \xrightarrow {t/G} Y/G$. Likewise, a $2$-morphism $f:A\to B$ is assigned: $f/G: A/G \to B/G$.

3.4 Totalizations of Burnside functors

Associated to any Burnside functor $\underline {2}^n \to \mathscr {B}_G$ is a chain complex, called the totalization (see [Reference Lawson, Lipshitz and SarkarLLS17a, Definition 5.1] and [Reference Sarkar, Scaduto and StoffregenSSS20, § 3.6]). For a set $X$, let $\mathcal {A}(X)$ denote the free abelian group generated by $X$. Let $X \xleftarrow {s} A \xrightarrow {t} Y$ be an equivariant correspondence. Define a map $\mathcal {A}(A) : \mathcal {A}(X) \to \mathcal {A}(Y)$ by

(3.1)\begin{equation} \mathcal{A}(A)(x) = \sum_{a\in s^{{-}1}(x)} t(a). \end{equation}

Let $F:\underline {2}^n \to \mathscr {B}_G$ be a Burnside functor. For $u\geq v$, let $A_{u,v}$ denote the correspondence assigned by $F$ to the morphism $\varphi _{u,v} : u \to v$. The complex $\text {Tot}(F)$ is defined by

\[ \text{Tot}(F) : = \bigoplus_{ u \in \underline{2}^n} \mathcal{A} (F(u)), \]

with the term $\mathcal {A}(F(u))$ in homological degree $\vert {u} \vert$. The differential

\[ \partial : \bigoplus_{\vert {u} \vert = i } \mathcal{A}(F(u)) \to \bigoplus_{\vert {v} \vert=i+1} \mathcal{A}(F(v)) \]

is given on summands by maps $\partial _{u,v} : \mathcal {A}(F(u)) \to \mathcal {A}(F(v))$, for $\vert {u} \vert = i$, $\vert {v} \vert = i+1$, defined as

\[ \partial_{u,v}(x) = ({-}1)^{s_{u,v}} \mathcal{A}(A_{u,v}). \]

The sign assignment $s_{u,v}$ ensures that $\partial ^2=0$ (see [Reference Bar-NatanBar05, § 2.7]; see also [Reference Lipshitz and SarkarLS14a, Definition 4.5] for a discussion of $s_{u,v}$).

If $X$ is a $G$-set, then $\mathcal {A}(X)$ is naturally a $\mathbb {Z}[G]$-module. Moreover, if $X \xleftarrow {s} A \xrightarrow {t} Y$ is an equivariant correspondence, then the map $\mathcal {A}(A)$ is $\mathbb {Z}[G]$-linear. Thus, if $F$ is an equivariant Burnside functor taking values in $\mathscr {B}_G$, then $\text {Tot}(F)$ is a complex of $\mathbb {Z}[G]$-modules.

3.5 Natural transformations of Burnside functors

There is a canonical identification $\underline {2}^{n+1} = \underline {2} \times \underline {2}^n$. A natural transformation $\eta : F_1\to F_0$ of Burnside functors $F_1, F_0: \underline {2}^n \to \mathscr {B}_G$ is a functor $\eta :\underline {2}^{n+1} \to \mathscr {B}_G$ such that the restriction of $\eta$ to $\{i\} \times \underline {2}^n$ is equal to $F_i$.

In the context of natural transformations, we will think of $\underline {2}^{n+1} = \underline {2} \times \underline {2}^n$ as two ‘horizontal’ copies of $\underline {2}^n$ with vertical edges connecting them, pointing downwards. The top copy of $\underline {2}^n$ corresponds to $\{1\} \times \underline {2}^n \subset \underline {2} \times \underline {2}^n$ and likewise the bottom copy corresponds to $\{0\}\times \underline {2}^n \subset \underline {2}\times \underline {2}^n$. Recall that for $u\geq v$, $\varphi _{u,v}$ denotes the unique element in $\text {Hom}_{\underline {2}^n}(u,v)$. We distinguish two types of morphisms in $\underline {2} \times \underline {2}^n$. First, for each $u\in \underline {2}^n$, there is a morphism

\[ (\varphi_{1,0}, \text{id}_u): (1,u) \to (0,u). \]

We denote this edge by $e_u$ and think of it as a vertical arrow

The second type consists of morphisms in $\underline {2}\times \underline {2}^n$ of the form

\[ (\text{id}_i,\varphi_{u,v}): (i,u) \to (i,v), \]

where $i \in \{0,1\}$ and $u,v\in \underline {2}^n$ with $u\geq v$. We will denote these morphisms by $\varphi ^i_{u,v}$, and think of them as living in the ‘horizontal’ cube $\{i\} \times \underline {2}^n$.

With these conventions, the diagram

(3.2)

lives in the horizontal cube $\{i\}\times \underline {2}^n$ and the diagram

(3.3)

is between the horizontal cubes. In the context of natural transformations, we will often not label some or all of the edges, with the understanding that the above conventions (

) and (

) are followed. To define a natural transformation $\eta :F_1\to F_0$, one needs to specify a correspondence $\eta (e_u)$ for each vertical edge $e_u$, a $2$-morphism $\eta _{u,v} : \eta (e_v) \circ \eta (\varphi ^1_{u,v}) \to \eta (\varphi ^0_{u,v}) \circ \eta (e_u)$ for each vertical face as in (

), and verify that the hexagon of Lemma 3.1 commutes.

Given a natural transformation $\eta : F_1\to F_0$, there is an induced map $\text {Tot}(\eta ) : \text {Tot}(F_1) \to \text {Tot}(F_0)$. The map $\text {Tot}(\eta )$ is defined on each summand by $\mathcal {A}(\eta (e_u)): \mathcal {A}(F_1(u)) \to \mathcal {A}(F_0(u))$.

3.6 A strategy for constructing natural isomorphisms

Given Burnside functors $F_1, F_0 : \underline {2}^n \to \mathscr {B}_G$, a natural isomorphism from $F_1$ to $F_0$ is a natural transformation $\eta : F_1\to F_0$ such that $\eta (e_u) : F_1(u) \to F_0(u)$ is an isomorphism in $\mathscr {B}_G$ for each vertex $u$. We will have several occasions to show that two Burnside functors are isomorphic (Propositions 4.3, 4.4, and 7.1). The general strategy is the same in all these cases, so we outline it here.

Note that a correspondence $X \xleftarrow {s} A \xrightarrow {t} Y$, thought of as a morphism in $\mathscr {B}_G$, is an isomorphism if and only if $s$ and $t$ are bijective. In particular, given an equivariant bijection $t: X\to Y$, the correspondence $X \xleftarrow {\text {id}} X \xrightarrow {t} Y$ is an isomorphism in $\mathscr {B}_G$, with inverse $Y \xleftarrow {t} X \xrightarrow {\text {id}} X$ (up to a canonical identification of a set with the diagonal in its cartesian square).

Suppose that we are given two functors $F_1, F_0 : \underline {2}^n \to \mathscr {B}_G$ and equivariant bijections $\psi _{u}: F_1(u) \to F_0(u)$ for each vertex $u \in \underline {2}^n$. For $u\geq _1 v$, let

\[ F_i(u) \xleftarrow{s^i_{u,v}} A_{u,v}^i \xrightarrow{t^i_{u,v}} F_i(v) \]

be the correspondence assigned to the edge $\varphi _{u,v}:u\to v$ by $F_i$ for $i=0,1$. Suppose also that for each $u\geq _1 v$, the following conditions hold.

1. (NI 1) $A^i_{u,v} \subset F_i(u) \times F_i(v)$, and the $G$-action on $A^i_{u,v}$ is inherited from the diagonal $G$-action on $F_i(u)\times F_i(v)$ (i.e. $g(x,y) = (g x, g y)$ for $g\in G$, $(x,y) \in F_i(u) \times F_i(v)$).

2. (NI 2) The map $F_1(u) \times F_1(v) \xrightarrow {\psi _u\times \psi _v} F_0(u) \times F_0(v)$ restricts to a bijection $A^1_{u,v} \to A^0_{u,v}$, denoted $\psi _{u,v}$.

3. (NI 3) The source and target maps, $s^i_{u,v}$ and $t^i_{u,v}$, are restrictions of the projections $F_i(u) \twoheadleftarrow F_i(u)\times F_i(v) \twoheadrightarrow F_i(v)$.

In this situation, we have a systematic method for building a natural isomorphism $\eta : F_1 \to F_0$ using Lemma 3.1 as follows. Define $\eta$ on objects by

\[ \eta(i,u) := F_i(u) \]

for $i\in \{0,1\}$ and $u\in \{0,1\}^n$. We then define $\eta$ on each vertical edge $e_u : (1,u) \to (0,u)$ by

\[ \eta(e_u) = \big( F_1(u) \xleftarrow{\text{id}} F_1(u) \xrightarrow{\psi_u} F_0(u) \big). \]