In the following we give an outline of the construction. We start with the Khovanov–Rozansky link homologies, which are categorifications of the $\mathfrak{gl}_{N}$ quantum link invariants of Reshetikhin–Turaev [RT90]. These link homologies take the shape of functors

which were constructed by Ehrig–Tubbenhauer–Wedrich in [ETW18] following earlier work on functoriality by Bar-Natan [BN05], Clark–Morrison–Walker [CMW09], and Blanchet [Bla10], and using technology developed by Robert–Wagner [RW20] and Rose–Wedrich [RW16] following Mackaay–Stošić–Vaz [MSV09], Lauda–Queffelec–Rose [LQR15], and Queffelec–Rose [QR16].

It is worth emphasizing that the functors $\mathrm{KhR}_{N}$ considered here are defined combinatorially and normalized to be sensitive to framing changes, in contrast to earlier incarnations of Khovanov–Rozansky homology. In the following, all links are oriented and framed and all link cobordisms are oriented.

The first step in our construction is to show that Khovanov–Rozansky homologies make sense as functorial invariants of links in $S^{3}$, rather than just in ${\mathbb{R}}^{3}$. From the point of view of link embeddings and link cobordisms, there is not much difference between these two cases. A generic link embedding will miss the point $\infty$ if we consider ${\mathbb{R}}^{3}=S^{3}\setminus\{\infty\}$ and a generic link cobordism embedded in $S^{3}\times{[0,1]}$ will miss $\{\infty\}\times{[0,1]}$. However, the analogous statement is no longer true for isotopies of link cobordisms. While link embeddings and their cobordisms can be represented by link diagrams in ${\mathbb{R}}^{2}$ and movies between them, there are additional isotopies of link cobordisms in $S^{3}\times{[0,1]}$, that do not exist in ${\mathbb{R}}^{3}\times{[0,1]}$. In addition to the standard Carter–Rieger–Saito movie moves [CS93, CRS97], a link homology theory that is functorial in $S^{3}$ additionally has to satisfy the so-called sweep-around move (1.1), which encodes a small isotopy of a sheet of link cobordism through $\infty\times{[0,1]}$. The central technical result that we prove in §3 is the following.

This move is significantly more complex than any of the Carter–Saito movie moves because it lacks any locality after the projection to ${\mathbb{R}}^{2}$, and thus has to be checked for any tangle $T$ with two endpoints. We do this in §3 and thereby also demonstrate how computable cobordism maps in Khovanov–Rozansky homology have become.

The main tool in constructing the 4-manifold invariants $\mathcal{S}^{N}_{0}$ is a family of 4-categories with sufficiently well-behaved duals. This is in analogy with the case of quantum invariants of 3-manifolds, which—in one way or another—all depend on a suitable 3-category, such as the ribbon category $\Rep(U_{q}(\mathfrak{gl}_{N}))$ of finite-dimensional representations of quantum $\mathfrak{gl}_{N}$.

In fact, the 4-categories we construct should be thought of as categorified representation categories¹¹ 1 These are related, but not identical, to categories of higher representations of categorified quantum $\mathfrak{gl}_{N}$. of quantum $\mathfrak{gl}_{N}$. They are defined to have unique 0- and 1-morphisms and

• •

  2-morphisms are indexed by finite sets of points in a disk,

• •

  3-morphisms are indexed by tangles in a ball,

• •

  4-morphisms between two tangles $T_{1}$ and $T_{2}$ are elements of the Khovanov–Rozansky homology $\mathrm{KhR}_{N}(T_{1}\sqcup\overline{T_{2}})$ of the link obtained by reflecting $T_{2}$ and gluing it with $T_{1}$ along their corresponding endpoints.

The various ways of composing $k$-morphisms are purely geometric for $k\leq 3$ and use certain cobordism maps between Khovanov–Rozansky homologies to define composition of $4$-morphisms. We give two constructions of such 4-categories, following the axioms of a disklike 4-category in §5 and of a braided monoidal 2-category in §6.

We invite the reader to use Khovanov–Rozansky link homology to build interesting examples of 4-categories following different axiomatizations, and to explore the appropriate incarnations of the sweep-around property in these settings.

The construction of the 4-manifold invariant $\mathcal{S}^{N}_{0}$ is most straightforward when using the setting of a disklike 4-category or the related notion of a lasagna algebra, a 4-dimensional analog of a planar algebra which we introduce in §5. Indeed, the bigraded abelian group $\mathcal{S}^{N}_{0}(W;L)$ is constructed as a skein module (inspired by the 3-dimensional analogs of Conway, Przytycki [Prz91] and Turaev [Tur91]) spanned by certain decorated surfaces in $W$ bounding $L$, which we call lasagna fillings, modulo skein relations imposed by the operad structure of the lasagna algebra.

More generally, there are bigraded abelian groups $\mathcal{S}^{N}_{i}(W;L)$ for $i\geq 1$ that arise as homology groups of the blob complex defined in [MW12] and can be thought of as higher derived analogs of $\mathcal{S}^{N}_{0}(W;L)$. In fact, the construction of the blob complex was motivated by the idea of using the $\mathcal{S}^{N}_{i}(W;L)$ as tools for computing $\mathcal{S}^{N}_{0}(W;L)$. We can think of $\mathcal{S}^{N}_{i}(W;L)$ as analogous to the $i$-th Hochschild homology $\mathrm{HH}_{i}$, where the input algebra of the Hochschild construction has been replaced by the 4-category derived from $\mathrm{KhR}_{N}$ and the implicit circle in the Hochschild construction has been replaced by the 4-manifold $W$. In particular, when $W$ is the standard 4-ball, so $L$ is a link in the 3-sphere, then $\mathcal{S}^{N}_{0}(B^{4};L)$ is isomorphic to the usual Khovanov–Rozansky homology of $L$, and for $i>0$ the abelian groups are zero.

The Khovanov–Rozansky 4-categories can be fed into the general machinery of [Wal, MW12] to produce fully extended $4{+}\epsilon$-dimensional TQFTs. One consequence of this is that the invariants $\mathcal{S}^{N}_{*}$ satisfy a gluing formula [MW12, Theorem 7.2.1] expressed in terms of a tensor product over a category associated to the gluing locus. In particular, we expect that $\mathcal{S}^{N}_{*}(B^{3}\times S^{1};\{2n\text{ points}\}\times S^{1})$ is related to the Hochschild homology of the $\mathfrak{gl}_{N}$ analog of Khovanov’s arc algebra. For applications of the latter to link homology see Rozansky [Roz10] and Willis [Wil21] $(N=2)$ and Gorsky–Hogancamp–Wedrich [GHW21] ($N=\infty$).

We would like to emphasize that $\mathcal{S}^{N}_{*}$ should be thought of as a categorified analog of the $3{+}\epsilon$-dimensional skein module TQFTs, see Walker [Wal], or the 3-dimensional layers of Crane–Yetter–Kauffman TQFTs [CKY97] at generic $q$, but not the $2{+}1$-dimensional Witten–Reshetikhin–Turaev TQFTs [Wit89, RT91].

Precise relationships along these lines and calculations based on the gluing formula appear in later papers [MN20, MWW2, HRW3].

Current constructions of Khovanov–Rozansky link homologies proceed via a functorial invariant of tangles and tangle cobordisms up to isotopy, taking values in the bounded homotopy category of an additive category; see §2. Our proof of Theorem 1.1 is stronger than necessary in the sense that it shows that a certain equivalent reformulation of the sweep-around move holds on the chain level (i.e. not just up to homotopy) provided the tangle $T$ is presented as a partial braid closure.

It is an open question whether the Khovanov–Rozansky homologies are truncations of homotopy-coherent versions with values in chain complexes over the same additive category. If this is indeed the case, then it is plausible that our method of proof would be suitable for an analog of Theorem 1.1 in this setting. Given a fully homotopy-coherent invariant of links in $S^{3}$, we could construct a disklike 4-category enriched in chain complexes (rather than abelian groups), and then use a homotopy colimit construction to extend this invariant to 4-manifolds [MW12, §7]. The result would be a well-defined-up-to-coherent-homotopy chain complex assigned to a 4-manifold $W$ and a boundary condition $L$. At the end of this process we could take homology of this chain complex to produce an abelian group. The invariants $\mathcal{S}^{N}_{*}(W;L)$ can be thought of as an approximation to the latter, given by taking homology (too) early in the construction. One would then expect the two theories to be related by a spectral sequence.

The results here hold for the ordinary Khovanov–Rozansky $\mathfrak{gl}_{N}$ link homologies as well as for their $\mathrm{GL(N)}$-equivariant and deformed versions [Lee05, Kho06, BNM06, Wu12, ETW18]. In the case of links in $S^{3}=\partial B^{4}$, the passage from the ordinary to deformed settings gives rise to spectral sequences that were studied in [Gor04, Ras15, Wu09, RW16]. Lobb and Wu [Lob09, Wu09], following pioneering work of Rasmussen [Ras10], showed that the associated filtrations for the generically deformed knot homologies in $S^{3}=\partial B^{4}$ contain lower bounds on the slice genus, i.e. the minimal genus of smooth surfaces in $B^{4}$ bounding the knot. Using such invariants, Freedman–Gompf–Morrison–Walker have outlined a strategy for testing counterexamples to the smooth 4-dimensional Poincaré conjecture [FGMW10]. One motivation for studying 4-manifold invariants from Khovanov–Rozansky homologies is that analogous spectral sequences might give rise to lower bounds on the genera of smooth surfaces in 4-manifolds $W^{4}$ bounding knots in $M^{3}=\partial W^{4}$.

There have been several proposed approaches to constructing homology theories for links in 3-manifolds, or 4-manifold invariants, which either intended to categorify $\mathfrak{sl}_{2}$ or $\mathfrak{gl}_{N}$ quantum invariants or to directly generalize Khovanov–Rozansky homology. These include

1. (1)

   categorifying Witten–Reshetikhin–Turaev invariants at roots of unity, see e.g. Khovanov [Kho16], Qi [Qi14], Elias–Qi [EQ16] and Qi–Sussan [QS17],

2. (2)

   using 2-representations of categorified quantum groups in the sense of Rouquier [Rou08] and Khovanov–Lauda [KL10] to construct a 4-category that can be integrated over 4-manifolds, see e.g. Webster [Web17] for categorified tensor products,

3. (3)

   categorifying skein algebras and 3-manifold skein modules, see Asaeda–Przytycki–Sikora [APS04] Thurston [Thu14] and Queffelec–Wedrich [QW18a, QW21], starting from the thickened annulus, see Grigsby–Licata–Wehrli [GLW18], Beliakova–Putyra–Wehrli [BPW19] and Queffelec–Rose [QR18], or connect sums of $S^{1}\times S^{2}$, see Rozansky [Roz10] and Willis [Wil21].

4. (4)

   giving a mathematically rigorous construction of the BPS spectra (“relative Gromov–Witten invariants”) proposed by Gukov–Putrov–Vafa [GPV17] and Gukov–Pei–Putrov–Vafa [GPPV20] based on Gukov–Schwarz–Vafa [GSV05], see e.g. Gukov–Manolescu [GM21] and Ekholm–Shende [ES19],

5. (5)

   extending Witten’s gauge-theoretic interpretation of Khovanov homology [Wit12] from ${\mathbb{R}}^{3}$ to other 3-manifolds, see also Taubes [Tau13, Tau18].

Comparing these approaches with the invariants defined here may be an interesting topic for further research. We expect a close relationship with approach (2) already at the level of 4-categories, and with approach (3) since it uses the same underlying combinatorics. The latter is especially appealing since (3) is, on the one hand, computationally well-developed for thickened surfaces, but, on the other hand, poses many open questions about the categorification of skein algebras and related quantum cluster algebras, onto which our invariants might shed new light.

While this article was under review, Manolescu–Neithalath [MN20] have shown that the values of $\mathcal{S}^{N}_{0}$ on 2-handlebodies can be computed from the Khovanov–Rozansky homology of cables of attaching links. The procedure takes the form of evaluating the Khovanov–Rozansky homology of the attaching link colored by a categorical Kirby color, a structure developed in the prototypical case $N=2$ by Hogancamp–Rose–Wedrich in [HRW3]. More generally, the values of $\mathcal{S}^{N}_{0}$ can be computed for general 4-manifolds from a handle decomposition, see Manolescu–Walker–Wedrich [MWW2].

The authors would like to thank Ian Agol, Chris Douglas, Mike Freedman, Marco Mackaay, Anton Mellit, Stephen Morgan, and Hoel Queffelec for helpful conversations. Scott Morrison was partially supported by Australian Research Council grants ‘Low dimensional categories’ DP160103479 and ‘Quantum symmetries’ FT170100019. Paul Wedrich was supported by Australian Research Council grants ‘Braid groups and higher representation theory’ DP140103821 and ‘Low dimensional categories’ DP160103479.

The category $\Rep(U_{q}(\mathfrak{gl}_{N}))$ of finite-dimensional $U_{q}(\mathfrak{gl}_{N})$-modules is a ribbon category and thus provides Reshetikhin–Turaev invariants of framed oriented tangles with components labeled by objects of $\Rep(U_{q}(\mathfrak{gl}_{N}))$. A framed oriented link $L$ labeled by the $U_{q}(\mathfrak{gl}_{N})$-module $V={\mathbb{C}}^{N}(q)$ yields an endomorphism of ${\mathbb{C}}(q)$, the tensor unit in $\Rep(U_{q}(\mathfrak{gl}_{N}))$, which is just multiplication by the $\mathfrak{gl}_{N}$ link polynomial of $L$.

While we will focus on invariants of links labeled by $V$, it is convenient to also consider the fundamental modules $\bigwedge^{k}(V)$ and their duals. Together, these generate the full monoidal subcategory $\mathrm{Fund}(U_{q}(\mathfrak{gl}_{N}))$, which admits a graphical presentation and which recovers $\Rep(U_{q}(\mathfrak{gl}_{N}))$ upon idempotent completion.

The ${\mathbb{C}}(q)$-linear pivotal category $\boldsymbol{\mathrm{Web}}_{N}$ has objects given by finite sets of points in an interval ${[0,1]}$, each labeled by an element of $\{n,n^{*}|n\in{\mathbb{Z}}_{>0}\}$. The morphisms are ${\mathbb{C}}(q)$-linear combinations of webs: oriented trivalent graphs, properly embedded in ${[0,1]}^{2}$, with edges labeled by a non-negative integer flow, considered up to isotopy relative to the boundary and certain local relations, including those shown in (2.1). The source and target of a web are determined by its intersections with ${[0,1]}\times\{0\}$ and ${[0,1]}\times\{1\}$, with downward oriented boundary points of label $n$ being recorded as $n^{*}$. Composition is given by the bilinear extension of stacking webs and the tensor product is given on objects by concatenating labeled intervals and on morphisms by the bilinear extension of placing webs side by side.

The morphisms in $\boldsymbol{\mathrm{Web}}_{N}$ are generated under composition, tensor product, and duality by identity morphisms and trivalent merge and split vertices:

The merge and split vertices encode the natural $U_{q}(\mathfrak{gl}_{N})$-intertwiners

respectively. The local relations in $\boldsymbol{\mathrm{Web}}_{N}$ include

together with the reflections of these relations in a vertical line. Edges labeled zero are to be erased and edges labeled by negative integers or integers greater than $N$ force the morphism to be the zero morphism. The local relations ensure that $\boldsymbol{\mathrm{Web}}_{N}\cong\mathrm{Fund}(U_{q}(\mathfrak{gl}_{N}))$ as ${\mathbb{C}}(q)$-linear pivotal categories, see [CKM14, QS19, TVW17]. In the following, we also consider an integral version of $\boldsymbol{\mathrm{Web}}_{N}$, which is defined over ${\mathbb{Z}}[q^{\pm 1}]$ and also satisfies the relations (2.1).

Foams provide a framework for a combinatorial description of Khovanov–Rozansky link homologies, in a similar way as webs are useful for the type A Reshetikhin-Turaev invariants. We will use $\mathfrak{gl}_{N}$-foams constructed via the combinatorial evaluation formula for closed foams due to Robert–Wagner [RW20]. More precisely, we will organize these $\mathfrak{gl}_{N}$-foams into a monoidal bicategory $\boldsymbol{\mathrm{Foam}}_{N}$ which categorifies the integral form of $\boldsymbol{\mathrm{Web}}_{N}$.

The graded, additive monoidal bicategory $\boldsymbol{\mathrm{Foam}}_{N}$ has objects given by finite sets of points in ${[0,1]}$, each labeled by an element of $\{n,n^{*}|n\in{\mathbb{Z}}_{>0}\}$. The 1-morphisms are (formal direct sums of formal grading shifts of) webs, properly embedded in ${[0,1]}^{2}$ and connecting boundary points of appropriate labels. Note that webs are not considered up to any relations in $\boldsymbol{\mathrm{Foam}}_{N}$. The 2-morphisms are (matrices of degree zero) ${\mathbb{Z}}$-linear combinations of $\mathfrak{gl}_{N}$-foams in ${[0,1]}^{3}$, considered up to isotopy relative to the boundary and certain local relations, as defined in [ETW18, Section 2]. The three compositions are given by (the bilinear extension of) stacking these topological objects along the three interval directions.

Foams are the natural notion of cobordisms between webs and the relations between $2$-morphisms in $\boldsymbol{\mathrm{Foam}}_{N}$ are chosen so that the defining web equalities in $\boldsymbol{\mathrm{Web}}_{N}$ can be lifted to explicit web isomorphisms in $\boldsymbol{\mathrm{Foam}}_{N}$. We refer to [ETW18] for a rigorous definition of $\mathfrak{gl}_{N}$-foams, as well as a complete description of the relations between them, and a survey of various flavors of $\boldsymbol{\mathrm{Foam}}_{N}$. Here we only comment on aspects relevant to the rest of this paper.

Foams are represented by 2-dimensional cell complexes, such that every point has a neighborhood either modelled on ${\mathbb{R}}^{2}$, three half-planes meeting in a line, or the cone on the 1-skeleton of a tetrahedron. Such cone points are called singular vertices of the foam. The points on the line in the second case form a seam of the foam, and the connected components of the set of manifold points are called the facets of the foam. An example of a foam with six singular vertices is shown in Figure 1. The facets are oriented and labeled by positive integers. If three facets meet along a seam, then two of their labels, say $a$ and $b$, sum to the third, $a+b$. The orientation of the seam agrees with the orientation induced by the $a$ and $b$ facets, and disagrees with the $a+b$ facet.

Each facet of a foam in $\boldsymbol{\mathrm{Foam}}_{N}$ admits an action of the algebra of symmetric functions $\Lambda$. This is to say that facets may be decorated by points labeled by symmetric functions, which are allowed to move freely on facets. A point labeled by a product $fg\in\Lambda$ may be split into two points labeled $f$ and $g$ respectively, and a foam with a point labeled $f+g\in\Lambda$ may be split into a sum of foams with points labeled $f$ and $g$ respectively. The $\Lambda$-actions on adjacent facets are compatible in the sense that $f\in\Lambda$ on an $a+b$ facet may be moved across a seam, where it distributes into $\Delta(f)\in\Lambda\otimes\Lambda$ acting on the adjacent $a$ and $b$ facets. The degree of a foam is computed as twice the degree of the symmetric function decoration, minus a weighted Euler characteristic, depending on facet labels.

$\boldsymbol{\mathrm{Foam}}_{N}$ is designed to have finite-dimensional spaces of $2$-morphisms, and in particular, the $\Lambda$-action on each $a$-facet factors through a finite-dimensional quotient, namely $H^{*}(\mathrm{Gr}({\mathbb{C}}^{a}\subset{\mathbb{C}}^{N}))$, the cohomology ring of the Grassmannian of $a$-dimensional subspaces of ${\mathbb{C}}^{N}$, which is obtained as quotient of $\Lambda$ by the ideal $\langle h_{N-a+i}|i>0\rangle$ generated by sufficiently large complete symmetric functions. In the case of a $1$-labeled facet, the symmetric function $e_{1}=h_{1}$ is called the dot.

The construction of Khovanov–Rozansky link homologies now proceeds in two steps. The first step is a functor that sends link diagrams to chain complexes in $\boldsymbol{\mathrm{Foam}}_{N}$ and link cobordisms to chain maps, which depend only on the isotopy type of the cobordism up to homotopy. The second step evaluates such a chain complex to a bigraded abelian group through a representable functor and taking homology.

In one direction, by forgetting the condition on the projection and ignoring the crossing order, this category is equivalent to the usual category of all embedded, framed oriented links and link cobordisms. In the other direction, the category ${\mathbb{R}}^{3}\boldsymbol{\mathrm{Link}}^{\circ}$ is equivalent to the category whose objects are link diagrams and whose morphisms are sequences of Reidemeister moves, Morse moves, planar isotopies, and formal reorderings, considered up to Carter–Rieger–Saito movie moves [CS93, CRS97].

We will now describe the construction of a functor $\left\llbracket-\right\rrbracket\colon{\mathbb{R}}^{3}\boldsymbol{\mathrm{Link}}^{\circ}\to\boldsymbol{\mathrm{K}}^{b}(\boldsymbol{\mathrm{Foam}}_{N})$, with target given by the bounded homotopy category of $\boldsymbol{\mathrm{Foam}}_{N}$. In particular, the functor sends link diagrams to certain bounded chain complexes of webs and foams. On single, $1$-labeled crossings, it is defined as:

The underlined term is placed in homological degree zero. We call the non-identity webs that appear here thick edges. The differentials in both complexes are given by the combinatorially simplest foam between the two shown webs. We call them unzip and zip foams respectively. The reader be warned that the assignments in (2.3) differ from the conventions in [KR08, Fig. 46] by a $q$-grading shift of magnitude $(N-1)w$, where $w$ denotes the writhe of the diagram; the latter further differ from the conventions in [ETW18, Equation (3.2)] by mirroring.

A link diagram with several crossings (in a specified order) is sent to the chain complex constructed from the formal tensor product of the crossing complexes (2.3) (in that order) by gluing its resolutions into the link diagram in place of the original crossings.

The chain complexes associated to link diagrams which differ only by Reidemeister moves are homotopy equivalent, see Sections 3.3–3.5. Similarly, one can define chain maps for Morse moves. However, a highly non-trivial fact is that there exists a coherent choice for such chain maps.

In fact, Theorem 2.4 holds in much greater generality, including colored links and the equivariant framework mentioned in Remark 2.2. More importantly for us, the theorem holds locally, i.e. for tangle diagrams and tangle cobordisms.

Given a braid word $\beta$ for a braid $[\beta]\in\mathrm{Br}_{n+1}$, we can get a 1-1-tangle diagram by taking the braid closure of the $n$ rightmost strands. We say that such 1-1-tangle diagrams are in almost braid closure form. From a 1-1-tangle diagram $T$, one can obtain link diagrams $L$ and $L^{\prime}$ by taking either the left- or right-handed closure of the single open strand. These diagrams are illustrated at the top left and top right of (3.1) respectively.

We note the following straightforward extension of the Alexander theorem.

Fix an almost closure $T$ of a braid word $\beta$ for $[\beta]\in\mathrm{Br}_{n+1}$. We call the right-hand closure $L$ and the left-hand closure $L^{\prime}$. We consider the following movies of intermediate diagrams and their associated chain maps between Khovanov–Rozansky complexes.

In the first row, the $\pm$ signs indicate the two versions of this movie, in which the horizontal strand passes in front of ($+$) or behind ($-$) $T$. We denote the composition along the top by $\mathrm{sw}_{+}$ and the composition along the bottom by $\mathrm{sw}_{-}$. In either case we first see a Reidemeister I move (denoted by $R1_{\pm}$), then a composite of Reidemeister II moves (denoted by $R2_{\pm}$), a number of Reidemeister III moves (each denoted by $R3_{\pm}$), a composite of inverse Reidemeister II moves ($R2^{-1}_{\pm}$), and finally an inverse Reidemeister I move ($R1^{-1}_{\pm}$). Our goal is to show that, after making careful use of the freedom, described later, to choose up-to-homotopy representatives of the chain maps for Reidemeister III moves, we have the following:

Together with Proposition 3.2, this will imply Theorem 1.1.

The proof of Theorem 3.3 will occupy the rest of this section.

We distinguish two types of crossings in the intermediate diagrams $L_{\pm}^{i}$. The crossings of the moving, horizontal, strand with everything else will be called external. The remaining crossings were already present in $T$ and will be called internal.

The braid word $\beta$ determines an ordering of the crossings in $T$, $L$, and $L^{\prime}$, namely from top to bottom. This ordering also induces an ordering of the internal crossings in all other diagrams in (3.1). The diagrams $L^{0}_{\pm}$ and $L^{x}_{\pm}$ have one additional external crossing. The diagrams $L^{i}_{\pm}$ for $1\leq i\leq x-1$ all have $2n+1$ external crossings, which are ordered from right to left. We will classify webs $W$ in each of these complexes according to the resolutions that appear at the crossings. For the following, let $M$ denote the number of crossings in $T$, $L$, and $L^{\prime}$.

The Reidemeister I chain maps are the following.

Here $\mathrm{cap}$ and $\mathrm{cup}$ simply denote the cap and cup foams, while $\mathrm{dcap}$ and $\mathrm{dcup}$ denote decorated cap and cup foams. The decoration is by the polynomial $\sum_{a+b=N-1}{X^{a}Y^{b}}$ where $X$ denotes the dot on the strand and $Y$ the dot on the circle; see (2.2). We have only assigned notation $R1_{\pm}$ and $R1^{-1}_{\pm}$ to those Reidemeister I chain maps that are relevant for the sweep-around move.

We will use Elias–Khovanov’s Soergel calculus [EK10a] to describe the chain maps associated to Reidemeister II and III moves. The Soergel calculus of type $A_{n-1}$ is a graphical incarnation of the 2-category of Soergel bimodules, which categorifies the Hecke algebra for $S_{n}$. For any $N\geq 2$, it admits a 2-functor to the monoidal subcategory of $\boldsymbol{\mathrm{Foam}}_{N}$ of webs and foams with $2n$ boundary components with suitable orientations, see e.g. [MV10]. Instead of describing these $2$-functors formally, we will just use the Soergel calculus as shorthand notation for foams using the following dictionary:

• •

  In the $A_{1}$ calculus, we have only a blue object, which we will interpret as the two strand web

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• •

  In the $A_{2}$ calculus, we have red and blue objects, interpreted as three strand webs

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• •

  Start dots and end dots (in any color) correspond to zip and unzip foams.

• •

  The trivalent vertices and correspond to digon creation and annihilation foams respectively. 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• •

  The 6-valent vertex corresponds to the foam shown in Figure 1.

The Reidemeister II chain maps are the following.

In both cases we have chosen to order the crossings from the top to the bottom. Now we can record two observations concerning the composite (inverse) Reidemeister II chain maps $R2_{\pm}$ and $R2^{-1}_{\pm}$.

In (3.1) we encounter four types of Reidemeister III moves. Namely, the moving strand can pass in front of or behind a positive or a negative crossing. In the following we show the front and back versions alongside each other. In every case, the moving strand is the one connecting the bottom left and top right boundary points.

In each variant of Reidemeister III, we order the crossings in each tangle from top to bottom. The parts of the complexes with internal homological degree zero—where the internal crossing is resolved in the parallel fashion—are highlighted in blue. The parts with internal homological degree $\pm 1$ are highlighted in yellow.

There is a 2-dimensional space of chain maps between the two sides of each Reidemeister III move [EK10b]. There is a 1-dimensional affine subspace of these chain maps which, given the previous choices for Reidemeister I and II maps, provides a functorial link invariant, by Theorem 2.4. (Note that their proof does not rely on any particular choice of chain maps from this subspace; any will do!) This subspace is characterised by the condition that the component of the chain map between parallel resolutions is the identity (this condition corresponds to the appearance of a blue highlighted $1$ in each chain map below). In the diagrams below, we parametrise this subspace by a variable $y$; shortly we shall specialize to $y=0$.

All choices of chain map in this affine subspace are homotopic, so for many purposes this structure can be ignored. For the present proof, however, it is quite important that we make the most convenient choice of up-to-homotopy representative.

When the moving strand passes a positive crossing we have: