Theorem 1.1. Suppose that we have a four-manifold with boundary , and let be the result of attaching a 3-handle to along a sphere . Let also be a framed link in disjoint from , and the corresponding link in . The equator of splits the sphere into two hemispheres, each of which induces a cobordism map from to . Then, the skein lasagna module is isomorphic to the coequalizer of these two cobordism maps. (See Theorem 3.7 for a more precise statement.)
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Skein lasagna modules and handle decompositionsThanks: CM was supported by NSF Grant DMS-2003488 and a Simons Investigator Award.Thanks: PW acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2121 “Quantum Universe” - 390833306.
Ciprian Manolescu
Address: Department of Mathematics, Stanford University, Stanford, CA 94305, USA
Email address: cm5@stanford.edu
,
Kevin Walker
Address: Microsoft Station Q, Santa Barbara, CA 93106, USA
Email address: kevin@canyon23.net
and
Paul Wedrich
Address: Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55,
20146 Hamburg, Germany
Email address: paul.wedrich@uni-hamburg.de
Original source: arXiv:2206.04616v2
Abstract.
The skein lasagna module is an extension of Khovanov–Rozansky homology to the setting of a four-manifold and a link in its boundary. This invariant plays the role of the Hilbert space of an associated fully extended -dimensional TQFT. We give a general procedure for expressing the skein lasagna module in terms of a handle decomposition for the four-manifold. We use this to calculate a few examples, and show that the skein lasagna module can sometimes be locally infinite dimensional.
1. Introduction
Homological invariants such as Khovanov homology [17] and Khovanov-Rozansky homology [21] are at the center of modern knot theory. These invariants were originally defined for links in . Extending them to links in arbitrary -manifolds is a problem that garnered much attention recently, from various perspectives (categorification at roots of unity [20, 9, 28], theoretical physics [34, 14, 13], etc.)
One such extension was introduced in [27], based on higher category theory and using the concept of blob homology [26]. Given a smooth, oriented, compact four-manifold and a framed oriented link in the boundary , the construction in [27] associates to the pair a homology theory graded by and denoted . One of the three integer gradings is called the blob degree, and for our purposes we will focus on the theory in blob degree zero, . This is called the skein lasagna module of and has a relatively simple definition, reminiscent of the definition of the skein module of a -manifold. The skein lasagna module is defined as the span of the lasagna fillings of with boundary , modulo an equivalence relation. The lasagna fillings are certain decorated surfaces connecting to other links in the boundaries of -balls inside , and the equivalences come from cobordism maps in Khovanov-Rozansky homology.
Skein lasagna modules are challenging to compute. It was proved in [27] that when , the invariant coincides with the Khovanov-Rozansky homology of the link . Further computational methods were developed in [25], with a focus on 2-handlebodies (four-manifolds obtained from by attaching 2-handles). This allowed the calculation of the skein lasagna modules (in some gradings) for four-manifolds such as the complex projective plane, and disk bundles over .
In this paper, building on the work in [27] and [25], we give a new formula for the skein lasagna module of a link in the boundary of an arbitrary four-manifold. We start by choosing a handle decomposition for the four-manifold. For simplicity, we may assume that we have a single 0-handle. We then study how the skein lasagna module changes under adding handles. Disjoint unions, 4-handles and many cases of 2-handles were already studied in [25], so the main thing left is to understand 1- and 3-handles.
With regard to 3-handles, we have the following:
Next, we combine Theorem 1.1 with the treatment of 2-handles in [25] to get a general result, reducing the calculation of the skein lasagna module to the case of 1-handles.
Recall that in [25], the skein lasagna module of a 2-handlebody was shown to be isomorphic to the so-called cabled Khovanov-Rozansky of the attaching link for the 2-handles; this is obtained from the Khovanov-Rozansky homologies of the cables of this attaching link, modulo certain cobordism relations. We define an analogue of the cabled Khovanov-Rozansky homology for two links in the boundary of (and, more generally, any other four-manifold); we call this the cabled skein lasagna module .
Theorem 1.2. Consider four-manifolds where
- •
is the union of one-handles;
- •
is obtained from by attaching two-handles along a framed link ;
- •
is obtained from by attaching three-handles along spheres ;
- •
be obtained from by attaching some four-handles.
Consider also a framed link , and view as a link in . Then, the skein lasagna module is isomorphic to the quotient of the cabled skein lasagna module by coequalizing relations coming from the 3-handles as in Theorem 1.1. (See Theorem 3.10 for a more precise statement.)
The cabled skein lasagna module is constructed from the invariants where is a family of framed links in consisting of and cables of the attaching link for the 2-handles. Thus, Theorem 1.2 allows us to express in terms of skein lasagna modules of links in (and maps between them).
The second half of our paper studies in more detail the skein lasagna modules for links in where . We work with coefficients in a field . By cutting along the cocores of the 1-handles, we reduce the problem of computing to a problem about skein lasagna modules for the (boundary of the) 0-handle with a family of framed links related to . For links in , the invariant is simply the Khovanov-Rozansky homology .
Theorem 1.3. Let with a nullhomologous link in the boundary that intersects the belt spheres of the 1-handles transversely in points for . Let denote the tangle obtained from by cutting open along the belt spheres. Then, the skein lasagna module is isomorphic to the quotient
where denotes a grading shift, and the relation is given by taking coinvariants for the actions of certain categories associated to the configurations of positively oriented and negatively oriented points in . (See Theorem 4.7 for a more precise statement.)
Furthermore, we will show that the isomorphisms from Theorem 1.3 are functorial in the following sense: They allow an expression of maps associated to cobordisms between links in in terms of components computed entirely from maps associated to link cobordisms in .
By combining Theorems 1.2 and 1.3 (plus the functoriality statement), we thus obtain a recipe for expressing the lasagna skein modules of any four-manifold in terms of Khovanov–Rozansky homologies of links in and maps associated to cobordisms in . The invariant is a quotient of a (typically infinite) direct sum of homologies of links by a subspace defined in terms of link cobordism maps.
Remark 1.4. Although the invariant for any four-manifold can be expressed purely in terms of link homology in , specifically , it would be difficult to prove directly that these expressions yield a four-manifold invariant. A direct proof of invariance, without comparing to the intrinsically defined invariants , would require checking handle slide and handle cancellation moves as well as higher coherence conditions between their composites. Handle slides for 2-handles are studied (for ) in [15] and instances of -handle cancellation are discussed in Example 3.8. Another interesting question concerns the behaviour of our algebraic decription of under reversing the handle decomposition of . However, our approach uses transversality arguments to isotope skeins away from cocores of handles to yield simplified handle formulas; hence, we do not expect these formulas to reflect the duality between - and -handles, because the duality does not respect cocores.
Specializing the setting of Theorem 1.3 to the case of a single 1-handle, we consider the link consisting of parallel circles, with of them oriented one way and the other way. We prove that is isomorphic to the zeroth Hochschild homology of the category . From here we get the following explicit calculation for .
Theorem 1.5. The skein lasagna module is
- (a)
one-dimensional when ;
- (b)
four-dimensional when ;
- (c)
infinite dimensional when .
Using methods analogous to those employed in part (a), we also show that is one-dimensional; see Corollary 4.2. For part (c), we actually show that is infinite dimensional in bidegree . This answers in the negative Question 1.7 from [25], about whether skein lasagna modules are always locally finite dimensional, i.e., finite dimensional in each fixed bidegree and homology class.
This still leaves open the following:
Question 1.6. If is simply connected, is always locally finite dimensional?
For , one can view the skein lasagna module as a variant of Khovanov homology for links in . Another version of Khovanov homology for these links was constructed by Rozansky (for ) in [31], and Willis [33] for arbitrary . The Rozansky–Willis homology is finitely generated in each bidegree and, thus, different from our theory. We expect that appears on the page of a spectral sequence converging to . See Section 4.6 for a further discussion and Section 4.7 for a conjectural extension of the Rozansky–Willis homology to links in the boundary of other four-manifolds.
Organization of the paper. In Section 2 we go over a few preliminaries about skein lasagna modules and Kirby diagrams. In Section 3 we study the behavior of skein lasagna modules under attaching 2- and 3-handles, proving Theorems 1.1 and 1.2. In Section 4 we focus on 1-handles, and prove Theorems 1.3 and 1.5.
Conventions. All the manifolds considered in this paper will be smooth, compact, and oriented. All links and surfaces are oriented and normally framed.
Acknowledgements. This paper builds on previous joint work and many enlightening conversations of KW and PW with Scott Morrison, without which this paper probably would not exist. We would also like to thank Matthew Hogancamp and Ikshu Neithalath for helpful comments on a draft of this paper.
2. Preliminaries
2.1. Skein lasagna modules
We start by reviewing the construction of skein lasagna modules from [27, Section 5.2].
Following [27] and [25], for a framed link , we write
for the version of Khovanov-Rozansky homology. Here, denotes the homological grading and denotes the quantum grading.
If we have an oriented manifold diffeomorphic to the standard -sphere , and a framed link , we can define a canonical invariant as in [27, Definition 4.12]. We sometimes drop from the notation and simply write .
Given a framed cobordism from to , there is an induced map
which is homogeneous of bidegree .
Let be a four-manifold and a framed link. A lasagna filling of with boundary consists of
- •
A finite collection of disjoint -balls (called input balls) embedded in the interior or ;
- •
A framed oriented surface properly embedded in , meeting in and meeting each in a link ; and
- •
for each , a homogeneous label
The bidegree of a lasagna filling is
If is a -ball, we can define a cobordism map
and an evaluation
We define the skein lasagna module as the bigraded abelian group
where is the transitive and linear closure of the following relation:
- (a)
Linear combinations of lasagna fillings are set to be multilinear in the labels ;
- (b)
Furthermore, two lasagna fillings and are set to be equivalent if has an input ball with label , and is obtained from by replacing with another lasagna filling of a -ball such that , followed by an isotopy rel (where the isotopy is allowed to move the input balls):
For future reference, here is a useful lemma.
Lemma 2.1. Let and be as above, and fix balls , one in each connected component of . Then, the equivalence relation defining can be alternatively be described as the transitive and linear closure of the following relation:
- •
Linear combinations of lasagna fillings are set to be multilinear in the labels ;
- •
Lasagna fillings that are isotopic rel are set to be equivalent;
- •
Two lasagna fillings are also set to be equivalent if they differ as in (b) above, where the input ball is one of the chosen balls .
Proof. If and are equivalent as in the lemma, let us show that they are equivalent as in the definition of the skein lasagna module. The only new relation is the isotopy, which can be thought of as a particular instance of (b), where is replaced by a slightly smaller ball with the same decoration (and is a product cobordism).
Conversely, if and are equivalent as in the definition of the skein lasagna module, we only have to consider the case when they are related by (b). We can then isotope to turn it into the ball in the same connected component, and view (b) as a combination of the moves in the lemma. ∎
Skein lasagna modules decompose according to relative homology classes, as noted in [25, Section 2.3]:
| (1) |
Observe that in the case where is not null-homologous in (i.e. ), then there are no lasagna fillings, so . When , consider the boundary map in the long exact sequence of the pair :
The only classes that can contribute non-trivially are those that map to the fundamental class under . Let us introduce the notation
Note that, using the long exact sequence of the pair, the difference of two classes in can be identified with an element of . Thus, is a torsor over ; it can be identified with the latter group after choosing a base element in .
2.2. Gluing and cobordisms
Let us consider two four-manifolds and that have some part of their boundaries in common, as follows:
where denotes disjoint union. We can glue and along to form a new four-manifold with boundary . Suppose we are also given links , and . Let denote the mirror reverse of . Then, we have a map
| (3) |
obtained by gluing lasagna fillings along :
It is easy to see that if two lasagna fillings and are equivalent in , and and are equivalent in , then and are equivalent in , so (3) is well-defined.
Starting from here, we see that skein lasagna modules are functorial under inclusions, in the following sense. We consider the case when , and we fix a lasagna filling of with boundary . We can think of as a cobordism from to . Then, there is an induced cobordism map
| (4) |
Observe that the maps (4) behave well with respect to compositions:
| (5) |
Furthermore, in terms of the decompositions (2), given , by attaching to it the class of in we get a class . Then, maps to . We let
| (6) |
denote the restriction of .
When the lasagna filling consists of a surface (an embedded cobordism from to ) with no input balls, we will simply write for . Furthermore, we could decorate with dots at a chosen location, for , as usual in foams; cf. [27, Example 2.3]. This corresponds to constructing a lasagna filling with input balls intersecting along unknots, each decorated with the generator
(This filling is equivalent to one where we consider a single input ball intersecting in an unknot, decorated with .) When the chosen location of the dot placement is clear from the context, then we denote the corresponding map by
| (7) |
2.3. Kirby diagrams
Let be a smooth, oriented, connected, compact four-manifold (possibly with boundary). By standard Morse theory, can be decomposed into -handles for , arranged according to their index . Furthermore, without loss of generality, we can arrange so that there is a unique 0-handle, and the number of 4-handles is either or , according to whether has empty boundary or not.
Denote the numbers of -, - and 3-handles by , and , respectively. After attaching the 1-handles to the 0-handle we get the handlebody , with boundary . (Here, denotes the boundary connected sum, and the usual interior connected sum.) The attaching circles for the 2-handles form a link
with components . The link also has a framing, which specifies how the 2-handles are attached. Once these are attached, the boundary of the resulting manifold must be of the form . Attaching the 3-handles gets rid of the summands of , so the resulting boundary is some -manifold . In the case , we stop here and we have . In the case where is closed, we must have and we attach the 4-handle (a four-ball) to at the last step to eliminate the boundary.
The handle decomposition allows us to represent by a Kirby diagram. This consists of drawing as pairs of spheres in , where we think of the spheres in each pair as identified to produce a 1-handle (and we also add the point at infinity to ). We then draw a picture of the attaching link for the 2-handles, where the link can go through the 1-handles. The framing of can be specified by drawing parallel copies of the components of . (The components that don’t go through the 1-handles can be viewed as living in ; for those, an alternative way to specify the framing is by an integer, which is the difference between the given framing and the Seifert framing.) To determine , in principle we should also specify the attaching spheres for the 3-handles. These are usually not drawn in the Kirby diagram. In the case where , this leaves no ambiguity, because there is a unique way to fill by 3-handles and then by a 4-handle.
For example, we show here a Kirby diagram of with one 1-handle and three 2-handles. For the attaching curve of the 2-handle that goes through the 1-handle, we specified the framing by drawing a parallel copy by a dashed curve; for the other 2-handles, we used numbers:
For more details about the subject, we refer to the book [10].
3. Two- and three-handles
3.1. Two-handles
The paper [25] contains a description of the skein lasagna module for 2-handlebodies (four-manifolds made of a 0-handle and some 2-handles), where the link is empty, or at least local (contained in a -ball). The description is in terms of the Khovanov-Rozansky homology of cables of the attaching link .
In this subsection we extend that description to the case where we attach 2-handles to any four-manifold , to obtain a new manifold . Moreover, we do not impose any restriction on the link . The formula is very similar to that in [25]. The role of the Khovanov-Rozansky homology will be played by the skein lasagna module , which can be thought of as a link homology for links in the boundary of . (When , we have .)
Let be the components of the framed link along which the 2-handles are attached. The framing gives diffeomorphisms between tubular neighborhoods of each and . Given -tuples of nonnegative integers
we let denote the framed, oriented cable of consisting of negatively oriented parallel strands to and positively oriented parallel strands. Here, the notion of parallelism for the strands is determined by the framing, that is,
for fixed points
After attaching 2-handles to along , we obtain the manifold . Suppose we are given a framed link . Generically, we can assume that stays away from the attaching regions of the 2-handles, and therefore we can represent it as a link in , disjoint from (but possibly linked with) . (There are various ways of isotoping off of the attaching regions; the results of the calculation will be isomorphic.) We let
be the union of and , where we do the cabling on the components of by choosing the tubular neighborhoods of to be disjoint from . (Note that is not a split disjoint union.)
We seek to express the skein lasagna module in terms of . To do this, we need to introduce a few more notions.
For each , let be the subgroup of the braid group on strands that consists of self-diffeomorphisms of rel boundary (modulo isotopy rel boundary) taking the set to itself and the set to itself. By taking the product with the identity on , a braid element induces a self-diffeomorphism of , which can be pulled back (via ) to a self-diffeomorphism of . This gives a group action
Let denote the basis vector. Two strands parallel to , if they have opposite orientations, co-bound a ribbon band in . By pushing into so that it is properly embedded there, and taking the disjoint union with the identity cobordisms on the other strands, we obtain an oriented cobordism (still denoted ) from to . For , we can decorate with dots, and obtain a cobordism map
which changes the bigrading by .
Next, recall that we have a decomposition (2) for the skein lasagna module , according to homology classes in . Let us see how these homology classes are related to the similar ones in . Consider the tubular neighborhood , which is a union of solid tori. Express as the union
where is the union of the new 2-handles, and is a connecting cylinder between and . Let also
We identify with and denote by (part of the boundary ).
The Mayer-Vietoris sequence for relative to the union of and reads
Observe that, by excision, . From here we obtain an exact sequence
| (8) |
Thus, an element in can be identified with its image in , which we write as a pair .
Let us further identify with by letting the th handle correspond to the coordinate vector . Then, we write
and let denote its positive part and its negative part; i.e., and . We also let .
Let and consider the cable . The fact that is in the kernel of the map to in (8) implies the existence of a (unique) class
which is sent to by the natural map to
From now on, using the deformation retraction from to , let us think of as a class in .
Definition 3.1. The cabled skein lasagna module of at level and in class is
where the equivalence is the transitive and linear closure of the relations
| (9) |
for all ; , and
Theorem 3.2. Let be a four-manifold and be a framed link. Let be obtained from by attaching 2-handles along a framed link disjoint from . Then, for each , we have an isomorphism
Proof. An element is represented by a linear combination of lasagna fillings in , where . We define to be the class of the linear combination of lasagna fillings with the same input data as , but with the surfaces given by attaching to each (along its boundary) the disjoint union of negatively oriented discs parallel to the core of 2-handle and positively oriented such discs (union over all ).
We also define a map in the opposite direction, as follows. Let be a lasagna filling in with surface . We isotope the input balls of to be inside , and isotope the surface such that its intersection with the 2-handles consists of several disks parallel to their cores. Removing these disks produces a lasagna filling of with boundary on a link of the form . We let this be .
The proofs that and are well-defined and inverse to each other are similar to the proof of Theorem 1.1 in [25], which dealt with the case and . The extension to arbitrary and is obtained by replacing the Khovanov-Rozansky homologies with the skein lasagna modules in . (In the formulation here, the proof of the statement is even slightly clearer since it relates lasagna skein modules with lasagna skein modules. In particular, we do not have to choose standard lasagna fillings with “slighly smaller input balls”, as these were only required when comparing with .) ∎
Remark 3.3. In some cases it is known that the braid group actions on the link homology of cabled links factor through the symmetric group. For Khovanov homology of links in , this was shown by Grigsby–Licata–Wehrli [12, Theorem 2]. For the homology of links in (or ) a similar argument works in the case of parallelly oriented strands [11, Section 6.1]. We have no reason to doubt that the same could be true for anti-parallel strands, i.e. in the situation relevant for , but we do not currently know how to prove it.
We will primarily be using the results from this subsection in the case where the role of is played by
a manifold obtained from a 0-handle by attaching some 1-handles. We denote by . Then, , so for any null-homologous , and the decomposition (2) for skein lasagna modules of links in is trivial (consists of a single summand). Moreover, in this case an element is uniquely determined by its image in . Indeed, the exact sequence
show that the component is determined by its image in
The part in has to be the fundamental class , while the part in is the image of under the isomorphisms
Therefore, in this case the class is redundant (being determined by ), so we simply drop it from the notation, writing for example instead of for the classes in . With this in mind, the isomorphism from Theorem 3.2 is written as
| (10) |
3.2. Three-handles
In [25, Proposition 2.1] the following result was shown:
Proposition 3.4. Let be the inclusion of a four-manifold into . Then we have a natural map
If is the result of a -handle attachment to , then is a surjection for and an isomorphism for .
Corollary 3.5. We have , concentrated in bidegree zero.
In this section we focus on the case of 3-handle attachments. We will generalize the statement of Proposition 3.4 to 3-handle attachments in the presence of boundary links and explicitly describe the kernel of the resulting maps on .
Consider the following setting. Let be a four-manifold with a framed link and an embedded -dimensional sphere , disjoint from . Let be the cobordism given by attaching a 3-handle to along , and let
Let be the outgoing boundary of , so that . Inside we have the two-dimensional annular cobordism , from to a new link . Given , let us consider the set of all whose equivalence class modulo is :
Remark 3.6. When (and therefore ), then is exactly the map from Proposition 3.4.
Let be the equator of (which is an unknot in ). Equip with an arbitrary orientation. By pushing a hemisphere of slightly from into the cylinder , and taking its union with , we obtain a properly embedded cobordism in , going from to . There are two such hemispheres, which produce two cobordisms, denoted and . We orient and so that their boundary orientation is the one on . (Note that they are therefore “oppositely oriented,” in the sense that they do not match up to produce an orientation on .) Let us identify with itself using a standard collar neighborhood. Then, the cobordism maps associated to and take the form
From here we get direct sum maps
and
Observe that these two maps have the same domain
and the same range Let
Theorem 3.7. The map associated to a 3-handle addition from to is surjective, and its kernel is exactly the image of . Therefore, is isomorphic to
that is, to the coequalizer of the maps and .
Proof. We first show that vanishes on the image of , that is,
Indeed, from the composition law (5) we see that the left hand side is associated to the surface cobordism and the right hand side to . However, inside the 3-handle , the sphere gets filled with a core , and therefore and are isotopic rel boundary. It follows that the two cobordism maps are the same.
Therefore, factors through a map
We need to prove that is bijective. For this, we construct its inverse . Given a lasagna filling of with boundary , observe that the cocore of the 3-handle is one-dimensional, and therefore we can isotope to be disjoint from this cocore; after this, we can push it into , to obtain a lasagna filling there, called , with boundary . We set
To see that is well-defined, we need to check that if two lasagna fillings and are equivalent in , then the corresponding fillings and differ (up to equivalences in ) by an element of . We use Lemma 2.1, in which we fix balls away from the 3-handle, and consider the equivalences listed in the lemma (with the ball replacements happening in ). Then, the equivalences in give rise to equivalences in , with one exception: an isotopy of the surfaces may intersect the one-dimensional cocore of (which is an interval). Generically, this happens in a finite set of points, each point at a different time during the isotopy. Every time the isotopy meets the cocore, the corresponding surfaces in differ by replacing a hemisphere of (with boundary some closed curve ) with its complement in . Up to an isotopy supported near , we can assume that is the equator with its chosen orientation. (For example, if is with the opposite orientation, we can rotate it by about a transverse axis to get with the original orientation.) Then, the hemispheres being interchanged are and and hence the classes of and differ by an element in the image of .
This shows that is well-defined, and its definition makes it clear that it is an inverse to . It follows that is bijective, and the conclusions follow. ∎
Example 3.8. Let and the sphere , where . Then attaching the 3-handle gives . Let us see what Theorem 3.7 gives in this case. For simplicity, we ignore the decomposition into relative homology classes.
The skein lasagna module of has the structure of a commutative algebra over , with the multiplication given by putting lasagna fillings side-by-side, in the decomposition
where is an interval. As a -algebra, was computed in [25, Theorem 1.2] to be
where comes from the lasagna filling corresponding to the closed surface , equipped with the standard orientation, and marked with dots. (As mentioned in Section 2.2, this is equivalent to introducing one input ball intersecting in an unknot labeled .)
The cobordism maps
are as follows. The unknot is contained in a ball in the boundary of (say, a neighborhood of the disk ). Then, according to [25, Corollary 1.5], we have
(Strictly speaking, Corollary 1.5 in [25] is phrased for coefficients in a field , due to the fact that its proof requires choosing a basis of . In our case, is the unknot, so is free over , and therefore the same argument applies with coefficients in .)
Both maps and correspond to capping the unknot by disks. The first map acts only on the factor and is given by
A useful picture to have in mind is that we can represent by a dotted disk (with the number of dots specified by the exponent of ), which is completed by to a dotted sphere that bounds a ball in , and hence can be evaluated to a scalar as shown above. To compute the action of , on the other hand, note that the disk completes the dotted disk to a homologically essential dotted sphere, corresponding to a generator in :
Therefore, taking the coequalizer of the two maps as in Theorem 3.7 boils down to setting
in . We deduce that
which is the known answer for the skein lasagna module of ; see [27, Example 4.6].
Remark 3.9. Example 3.8 gives an alternate formula for 3-handle attachments. Let us go back to the general setting in this section, with a 3-handle attached to an arbitrary four-manifold along a sphere to produce , and a framed link away from . Observe that is naturally a module over the algebra , with the module action being given by attaching fillings in a neighborhood of the sphere . It follows from the definitions that
Here, the algebra is the free polynomial ring in and is as a module over that algebra, where acts by and the other by . We conclude that
3.3. Handle decompositions
Let us now specialize the addition of 3-handles to the case where the initial manifold is a union of -, - and 2-handles. We will then have available to us the description of from Section 3.1.
If we attach a 3-handle to , in terms of Kirby calculus, the attaching sphere can be represented as a surface (of genus , and disjoint from ) with boundary some copies of the ’s (the attaching circles for 2-handles). Then is the union of and (parallel copies of) cores of the 2-handles.
We draw as a small unknot away from all , and let be the small disk it bounds. The other hemisphere is the complement of in , and goes over some of the handles. We let
This is a surface on whose boundary is the union of and several copies of the ’s. Let be the number of copies of in that appear with the negative orientation, and the number of those with the positive orientation. We form the vectors
We proceed to describe the maps and in this case. By Theorem 3.2 with notation as in (10), the range of these maps is identified with the direct sum of cabled skein lasagna modules . Similarly, their domain is identified with
We used here the fact that is split disjoint from all the attaching links for the 2-handles, and therefore each summand that appears in the definition of splits off a factor; moreover, the equivalence relation is compatible with this splitting.
The map is now easy to describe. It is induced by capping with a disk, so it only affects the factor , in a standard way. Precisely, we have
| (11) |
for all .
To describe the second map , consider the diagram
| (12) |
Here, in the top row we wrote for a pair as in Definition 3.1. The vertical maps from the first to the second row are induced by the inclusion of the summands into the cabled skein lasagna module; cf. Definition 3.1. The vertical maps from the second to the third row are the isomorphisms from Theorem 3.2.
Ignoring the middle dashed arrow for the moment, note that the above diagram commutes. Indeed, by the definition of in the proof of Theorem 3.2, the vertical compositions (from the first to the third row) are given by attaching cores of the 2-handles to lasagna fillings in . Note that we are attaching more cores on the right; namely, those in the boundary of , counted by the vectors and . The horizontal cobordism maps (as defined in Section 2.2) are given by attaching the surface (in the top row) and (in the bottom row). Because is the union of and the extra cores of 2-handles counted by and , the diagram (12) commutes.
Since the bottom vertical arrows in the diagram are isomorphisms, let us now add the middle dashed arrow, given by the map
Because (12) commutes, we deduce that this map is induced on the skein lasagna modules by applying the cobordism maps on each summand; this justifies the notation.
Recall that is the complement of the disk inside . Thus, we can write the cobordism maps in terms of the maps associated to the surface with dots, as in (7):
Fixing , the maps on various summands in the construction of the skein lasagna module induce a map:
such that
| (13) |
We are now ready to give a general formula for the skein lasagna module of a four-manifold decomposed into handles in terms of skein lasagna modules of 1-handlebodies. We will phrase it for an arbitrary number of handles.
Theorem 3.10. Consider four-manifolds where
- •
is the union of 1-handles;
- •
is obtained from by attaching two-handles along a framed link ;
- •
is obtained from by attaching three-handles along spheres ;
- •
is obtained from by attaching some four-handles.
Consider also a framed link . We represent by a Kirby diagram, viewing as a link in , and the spheres in terms of surfaces on with consisting of some copies of various components of (so that is the union of and the corresponding cores of the 2-handles).
Given
let be the set of all whose equivalence class modulo is .
Then, the skein lasagna module is isomorphic to the quotient of the direct sum of cabled skein lasagna modules by the relations
| (14) |
and
| (15) |
for all and .
Proof. First, note that the addition of 4-handles does not affect the skein lasagna module, in view of Proposition 3.4. Thus, we can consider instead of .
The skein lasagna module of viewed in the boundary of is given by according to Theorem 3.2. When we add a 3-handle, we divide by the relations
| (16) |
as proved in Theorem 3.7. In terms of the identifications from Theorem 3.2, the left hand side of (16) is given by Equation (11), and the right hand side by Equation (13). We thus get relations of the form (14) and (15). The generalization to multiple 3-handles is straightforward. ∎
Theorem 3.10 gives a description of an arbitrary skein lasagna module in terms of skein lasagna modules for links in the boundary of , and cobordism maps for surfaces in . In the next section we will obtain a further reduction to links in and cobordism maps between them, under the additional constraint of working with field coefficients; see Theorem 4.7.
4. One-handles
Consider four-manifolds and , where is the result of attaching a finite number of 1-handles to . The boundary of the cocore of each 1-handle is a -dimensional sphere that generically intersects links in a finite set of points. In this section we aim to compute in terms of the invariants of the four-manifold and some links related to .
Throughout this section we will work with coefficients in a field . Under this assumption is strictly monoidal under disjoint union (without terms) and sends mirror links to dual link homologies (without terms). As a consequence, is monoidal under (boundary) connect sum; see [25, Theorem 1.4 and Corollary 7.3]. We leave the investigation of the behavior under more general coefficient rings to future work.
4.1. One-handles away from links
We first consider the case when is disjoint from the cocores of the 1-handles. Up to a small isotopy, we may even assume that is disjoint from the entire boundary of the added 1-handles, i.e. that . As in Proposition 3.4, the corresponding invariants are related by a canonical map and we have:
Lemma 4.1. The inclusion induces an isomorphism
Proof. The proof is a straightforward generalization of the proof of [25, Theorem 1.4], which deals with boundary connected sums. The map is induced by the map sending lasagna fillings of to lasagna fillings of along the embedding . The inverse is given on lasagna fillings in by looking at their intersection with a neighborhood of the cocores of all 1-handles. Up to a small isotopy, each such intersection is an identity cobordism on a link . The inverse map is given by replacing it by a sum of pairs of input balls, labelled by basis and dual basis elements of respectively. The resulting linear combination of fillings can be isotoped into , and is equivalent to the original filling according to the neck-cutting lemma (Lemma 7.2 in [25]). ∎
Corollary 4.2. There are canonical isomorphisms
each sending to the respective empty lasagna filling.
Proof. The first isomorphism is given by a 1-handle attachment to as in Lemma 4.1. The second isomorphism can be proved similarly: Let be a lasagna filling of and consider its intersection with a fiber . Up to a small isotopy, we may assume that the filling intersects transversely (in lasagna sheet, not in input balls) and disjointly from . Then for small , the intersection is an identity cobordism on a link . We replace this by a sum over pairs of input balls labelled with basis and dual basis elements of respectively. The resulting closed lasagna filling is supported in a single and can, thus, be identified with a scalar multiple of the empty filling. ∎
Remark 4.3. It is instructive to evaluate the inverse to the canonical isomorphisms from Corollary 4.2 on surfaces of revolution generated by links. Any framed, oriented link or defines a vegetarian11 1 A lasagna filling consisting only of a surface, without input meat balls. lasagna filling of , which evaluates to a scalar multiple of the empty lasagna filling. It follows from the proofs of Lemma 4.1 and Corollary 4.2 that this scalar is the trace of the identity map on . Here it is important to take the Koszul signs in the symmetric monoidal structure on (homologically and quantum) bigraded vector spaces into account. The trace is thus , i.e. the quantum link polynomial of , specialized at . More generally, any endocobordism of defines a lasagna filling of that is a multiple of the empty filling, with coefficient given by the graded trace of the induced endomorphism of ; see e.g. [16, Section 6], [3, Section 10.1], [7, Theorem D] for related discussions of Lefschetz traces in the case of Khovanov homology.
4.2. Cutting and gluing 1-handles
Consider the process of cutting a lasagna filling of with boundary along the cocores of the 1-handles for . Let us assume that the lasagna sheet of intersects the cocores transversely in tangles . In particular, the link intersects the belt spheres geometrically in points, the boundary points of the tangle . The algebraic intersection numbers are all zero, since is null-homologous, as witnessed by . In this way, we obtain a lasagna filling of with boundary link
The latter is obtained by cutting open at the -tuples of boundary points and inserting copies of the tangles and , schematically:
Of course, the procedure of cutting lasagna fillings does not describe a well-defined map on the level of since it does not respect the skein relations. Instead we consider the reverse operation.
The process of gluing a lasagna filling works as follows. Let be a lasagna filling of with boundary link as above; i.e., inside we have pairs of embedded 3-balls , such that and for . Denote the numbers of boundary points by . Now we attach 1-handles with core-parallel lasagna sheets along the to obtain a lasagna filling of with boundary . Since the relations in are local, this induces a map:
| (17) |
The grading shift is there to compensate the change in Euler characteristic of the surfaces in lasagna fillings upon gluing.
Lemma 4.4. For every lasagna filling of , there exists a framed , such that is contained in the image of .
Proof. By a small isotopy, we may assume that satisfies the assumption of the cutting procedure described above. The statement now follows since cutting, albeit ill-defined, is manifestly a right-inverse to gluing. ∎
It follows that the gluing maps from (17) assemble to a surjective map from a direct sum of shifts of to . Here, the sum is indexed by all ways of writing as a contraction of links obtained by drilling out pairs of tangles and resealing the boundary points across the 1-handles. It remains to describe the kernel.
Definition 4.5. For fix a configuration of framed points in , partitioned into two halves with opposite co-orientations. We define a category enriched in bigraded -vector spaces with:
- •
objects: framed, oriented tangles in inducing the given orientation on
- •
morphisms given by
(18) (19)
with (grading-preserving) composition maps induced in the case of the right-hand side of (18) by the action of merging cobordisms, as described in [27, Section 6.1 (vertical composition of 2-morphisms)], and in the case of (19) induced by the gluing of lasagna fillings of balls.
Lemma 4.6. Let be a smooth, oriented, connected, compact four-manifold. Fix and consider a link that intersects in a tangle with boundary , i.e. . Now let be another such tangle and , then we have a grading-preserving gluing map
Moreover, these gluing maps are compatible with composition in in the sense that all diagrams of the following type commute:
Proof. Straightforward on the level of lasagna fillings. The map descends to the quotient since skein relations are local. ∎
The statement of Lemma 4.6 can be paraphrased as: the choice of a -ball with point configuration in equips with the structure of a bigraded module for the category . (Here the direct sum is taken over all links that intersect the boundary of the chosen -ball in the fixed configuration .)
Theorem 4.7. Let with a nullhomologous link in the boundary that intersects the belt spheres of the 1-handles transversely in points for . Let denote the tangle obtained from by cutting open along the belt spheres. Then we have an isomorphism:
where the relation is given by taking coinvariants for the actions of , i.e. by identifying the images of the actions
for all pairs of tangles , with boundary . (Here we have omitted a global grading shift.)
Proof. The map is defined by first considering the direct sum of the gluing morphisms
from (17). The coinvariants for the actions of clearly lie in the kernel, so we get an induced map from the indicated quotient to , which we again call the gluing map. It is surjective by Lemma 4.4, so it remains to prove injectivity.
Let be two equivalent linear combinations of lasagna fillings in , and let be respective preimages under the gluing map. We want to show that and are equivalent. Without loss of generality, we may assume that and are individual lasagna fillings (rather than linear combinations) and that they differ by a single move as in Lemma 2.1 with the relevant input ball fixed and disjoint from the cocores of the 1-handles in . If and differ by a replacement inside the fixed input ball or an isotopy supported away from the cocores, then and are equal in . If and differ by an isotopy supported in a neighborhood of the cocores, then or differ by an element of the subspace factored out. Since every isotopy of lasagna fillings can be factored in this way, we get that and are equivalent. ∎
Theorem 4.7 can also be summarized by saying that is computed by the zeroth Hochschild homology of a tensor product of -ball categories, namely one for each handle, with coefficients in a bimodule associated to the tangle that results from by cutting open along the belt spheres. We will discuss the details of this perspective in a special case in Section 4.3.
Remark 4.8. Similarly to the 2-handle formula from Theorem 3.2, the 1-handle formula from Theorem 4.7 expresses the skein module of the more complicated manifold as a quotient of a (countable) direct sum of invariants of simpler manifolds. A possibly relevant difference, however, is that the 2-handle formula features only finitely many summands with a given shift in quantum grading, whereas this number is infinite for the 1-handle formula.
The skein modules that have been computed using only the 2-handle formula, first and foremost in [25], are locally finite-dimensional, i.e. finite-dimensional in each bidegree. It is an open question whether this is true for all four-manifolds admitting handle decompositions without 1-handles. In the rest of this paper we will see that local finite-dimensionality may fail when 1-handles are present.
Finally we comment on the functoriality of the 1-handle formula from Theorem 4.7. We have seen that for is a colimit of link homologies for links in , which result from cutting along belt spheres and inserting pairs of tangles. Now consider a link cobordism from to where . We claim that the induced map
can also be expressed in terms of cobordism maps between links in . Recall that the cobordism map sends a lasagna filling of to the composite lasagna filling of . In a generic situation, cutting the cocores has the following local model. Here we display the filling in the inner tube and in the outer, spherical shell.
Let denote the tangle in that occurs as the intersection of with the th cocore and the tangle obtained from by cutting open along the belt spheres of . Denote by the number of outer boundary point of . Then the cobordism obtained from by cutting along the annuli, which are the intersection of with the cocores of 1-handles in , induces a cobordism map:
We claim that these components describe in terms of the colimit formulas (left-hand sides) from Theorem 4.7. To see this we first observe that the unequal grading shifts guarantee that the components have the same degree as (we have and is glued to along interval segments). Next we observe that after composing with the projection-inclusion into the colimit formula for , the resulting map no longer depends on the chosen location of cocores to cut. Moreover, the subspace factored out in the colimit formula for is annihilated by the map thus defined. Thus the components described above define a map , and by construction this agrees with .
4.3. Algebraic description of the 3-ball categories and their Hochschild homologies
Recall the following definition, from e.g. [4].
Definition 4.9. Let be a commutative ring and be a (small) -linear category. Then the zeroth Hochschild homology of , also called the trace of , is defined as the -module
where the spanning set for the subspace to be divided out is constructed from all pairs of cyclically composable morphisms, i.e. and for some .
If as in Definition 4.9 is not just enriched in -modules, but -graded -modules for some monoid , then inherits the structure of an -graded -module. The following is now an immediate consequence of Theorem 4.7 and the Definitions 4.5 and 4.9.
Corollary 4.10. Let and consider the link consisting of parallel circles with balanced orientations (that is, with circles oriented one way and the other way). Then, we have an isomorphism of bigraded -vector spaces:
| (20) |
We now recall some facts about the zeroth Hochschild homology, which we will use to show that the 1-handle formula may compute vector spaces which are not locally finite-dimensional.
Fact 4.11. Any functor of -linear categories induces natural -module homomorphism sending . This is well-defined since . If is an equivalence, then is an isomorphism; see e.g. [4].
Fact 4.12. Let and denote the canonical embeddings of into its additive and its idempotent completion, respectively. Then and are isomorphisms; see e.g. [4, Sections 3.4 and 3.5].
In a slight reformulation of the functoriality results from [8], the tangle invariant underlying the link homology over can be described as a 2-functor:
We now briefly explain the relevant algebraic structures here.
- •
As in [27, Definition 6.1] one defines a category of tangle diagrams, whose objects are finite words in the alphabet (which encode possible sequences of oriented boundary points for tangles) and whose morphisms are finite words in generating morphisms (where the index specifies the strands participating in the generator), that are admissible in the sense that the composite describes a tangle diagram. The composition is concatenation of words. For details see [27, Definition 6.1].
- •
is a -category whose objects and -morphisms are as in . The -morphisms are the framed, oriented tangle cobordisms in between standard lifts of tangle diagrams to actual tangles in , considered up to isotopy rel boundary.
- •
is a (monoidal) -category, enriched at the level of -morphism spaces in -vector spaces and equipped with grading shift functors on -morphisms. It has the same objects22 2 More generally, one can consider labelled oriented points as objects in , but we will not need labels other than . as . The -morphisms are (formal direct sums of grading shifts of) webs embedded in and the -morphisms are (matrices with entries given by) -linear combinations of foams embedded in , modulo certain local relations. For details see [8].
- •
is the (monoidal) -category that is obtained from by replacing its -linear -categories by the corresponding dg categories. This means it has the same objects, but the -morphisms are now chain complexes formed from -morphisms in , where the differentials are given by -morphisms in . The -morphisms spaces are chain complexes of homologically homogeneous and quantum grading-preserving maps, spanned by -morphisms from (not necessarily chain maps). The differential on -morphisms is the usual supercommutator with respect to the differential on the source- and target complexes. With respect to this differential the zero cycles are exactly the classical chain maps. There is also an enriched -hom in , which is assembled from -homs between objects shifted in quantum grading.
- •
is the cohomology category of . It has the same objects and -morphisms, but the -morphism spaces are now graded -modules obtained by taking cohomology. The zeroth cohomology is also called the homotopy category; its -morphisms are chain maps up to homotopy.
In the following we will also consider enriched -homs. For objects and -morphisms we define the bigraded -modules:
(21) Here one grading, the quantum grading, is given by the displayed direct sum, while the other grading, the homological grading, is already internal to . Using the grading shift automorphisms, these enriched -homs admit composition maps and thus assemble into a bigraded -linear enriched morphism category whose objects are the -morphisms from to .
- •
The functor is the identity on objects. On -morphisms it sends a tangle diagram to a chain complex of webs and foams in the way that is usual for link homology, and -morphisms, i.e. isotopy classes of tangle cobordisms are sent to the corresponding homotopy classes of chain maps as specified in the functoriality proof in [8].
We recall from [27, Section 6] that the tangle invariant corresponding to the link homology can be organized into a braided monoidal -category. Here we give a similar construction of this category (which was denoted in[27]) by replacing the top morphism layer of :
- •
objects are sequences of tangle endpoints, as in and ,
- •
1-morphisms consist of Morse data for tangles, as in and ,
- •
2-morphisms between tangles and with equal source and target objects are the bigraded -modules computed as the enriched 2-hom from (21) between the chain complexes of the tangles.
As an important special case, one gets for a framed, oriented link :
Moreover, if and are framed, oriented tangles with endpoints identified, so that we can form the link , then we set and have:
Given a -ball with a set of framed, co-oriented points in the boundary, together with a suitable identification of with , we associate to it the morphism category , whose objects are tangles from to . By construction, is equivalent to from Definition 4.5. Moreover, can be considered as a full subcategory of the bigraded enriched morphism category .
4.4. The 3-ball category with two points
Here we consider the categories from Section 4.3 in the special case when the source and target objects consist of a single point . In this case, the corresponding morphism category in is known to be equivalent to the dg category of complexes of free graded -modules; see e.g. [32, Lemma 3.35] for an argument in an equivalent setting. We record this equivalence and its consequence on the level of homology:
Here refers to the category of finitely-generated graded free -modules and refers to the dg category of bounded chain complexes over an additive category . Again we will use a superscript to refer to the corresponding enriched morphism spaces, computed via the ordinary morphism spaces between shifts of objects as in (21).
Now we specialize to and classify the indecomposable objects. Setting , the isomorphism classes of indecomposable objects (up to shifts in quantum and homological degrees) in are of the form:
for ; see [19, Section 3].
Next we compute the zeroth Hochschild homology of . In principle, there are two possible versions: using the ordinary or the enriched hom; see [5, Section 2.4]. In the case of the ordinary hom, we would obtain a -graded (namely homologically graded) -module, where records the action of the auto-equivalence provided by the shift in quantum grading. We will, however, use the enriched hom (indicated by the superscript ) to consider the morphism spaces as bigraded. In doing so, one obtains translation isomorphisms, which identify an object with all its gradings shifts. More specifically, between an object and its shift, the identity now represents an isomorphism of degree specified by the shift. The zeroth Hochschild homology of the resulting category carries the structure of a bigraded -vector space, since the endomorphism now acts as the identity.
Proposition 4.14. The bigraded zeroth Hochschild homology of has a basis given by the trace classes and for all . The identity morphisms on the complexes for are self-explanatory and their trace classes have bidegree . The endomorphism is a special case of a larger family of endomorphisms for of the following form:
where the only non-zero component is at (which may coincide with if ). The trace class of the morphism has bidegree . ( stands for shift right and apply .)
Proof. We abbreviate . Let denote the full subcategory generated by the indecomposable objects . By Fact 4.11 and the discussion of the beginning of the section, it suffices to compute the bigraded zeroth Hochschild homology of . To this end, we study closed homogeneous endomorphisms of the objects and trace relations between them.
We note that the components of a chain map between shifts of such objects can have quantum degree zero or two (a scalar multiple of or ). Since the differential in every complex is of quantum degree two, this means that closed morphisms with components of quantum degree zero are homotopic if and only if they are equal.
First we investigate the chain maps between shifts of objects with components of quantum degree zero. For positive homological shifts (right shift) there are simply no closed morphisms, i.e. no chain maps. In shift zero we have the identity on every (which does not factor through any with ) and for negative homological shifts we have closed maps that factor into a composite of closed maps through a shift of a with (by induction, one can show that their trace classes actually vanish). Thus in bidegree we have a basis of trace classes for .
Second we are interested in chain maps between shifts of objects with components of quantum degree two. In negative homological shifts (left shift) all such maps are nullhomotopic. In non-negative homological shift, every such map is homotopic to a scalar multiple of . However, one easily checks that the trace class of equals the trace class of . Since these have bidegree in the enriched of , we see that they are linearly independent. ∎
Note that the bigraded zeroth Hochschild homology of is not locally finite-dimensional! It is of countable dimension in bidegree with a basis given by for . Nevertheless, we have:
Proposition 4.15. The bigraded vector spaces
are four-dimensional, and in particular, locally finite-dimensional.
Proof. We have already explained the two isomorphisms. We now need to understand the essential image of under the full embedding into . We claim that the invariant of any -tangle decomposes into (shifts of) the indecomposable summands and , but never for . Provided this claim holds, we can compute as the Hochschild homology of the full additive subcategory of generated by and , and this again is isomorphic to the Hochschild homology of the full subcategory on the two objects and . Here we use that the zeroth Hochschild homology is preserved under proceeding to the additive and idempotent completion; see Fact 4.12. Following the same arguments as in Proposition 4.14, we see that it is -dimensional, spanned by and for
The key idea to prove the claim is that all complexes appearing in Khovanov homology come from complexes over by setting (though certainly not all complexes over have this property). Indeed, one can use equivariant Khovanov homology, defined over the ring to simplify the complex of a -tangle into a complex of graded free -modules. These decompose, up to homotopy equivalence and shift, into chain complexes of the form
Upon reducing to the ordinary Khovanov theory by tensoring with over , these complexes decompose into (shifts of) copies of and . ∎
Remark 4.16. A strong version of the so-called knight move conjecture posited that the complex of any long knot decomposes (up to homotopy equivalence) into one shifted copy of and some number of copies of ; see [19, Conjecture 1]. The argument in the previous proof shows that this can fail only due to the presence of more than one shifted copy of . Three copies of can be detected in the counterexample to the knight move conjecture found by Manolescu–Marengon [24].
Remark 4.17. One can also consider analogs of the skein modules based on equivariant or deformed versions of homology. For example, in one common choice for one works over . We can also try to compute the bigraded zeroth Hochschild homology of the 3-ball category with two points and of its ambient category in this setting. We have already listed the indecomposable of the latter above: the chain complexes . For the enriched isomorphism algebra of the complex is isomorphic to where is of bidegree . The trace classes of and its multiples are zero. Moreover, the trace class of is zero for every . This leaves the trace classes of the identities of for and the trace class of as linearly independent — the zeroth Hochschild homology is not locally finite-dimensional. However, it is currently not known which appear in complexes of -tangles. A copy of appears in [24].
4.5. The 3-ball category with four or more points
We claim that the -ball categories with points have zeroth Hochschild homologies that are no longer locally finite-dimensional. Again we restrict to the case of and work over a perfect field . Our strategy is to give a lower bound for the dimension of the zeroth Hochschild homology in terms of the split Grothendieck group. We briefly recall the relevant notions and results.
Definition 4.18. Let be an additive category. The split Grothendieck group of is defined as:
Definition 4.19. A -linear additive category is called Krull–Schmidt if every object decomposes uniquely into a finite direct sum of indecomposable objects with local endomorphism rings.
The following is clear from the definition:
Proposition 4.20. For a Krull-Schmidt category, the split Grothendieck group is a free abelian group on the isomorphism classes of indecomposable objects in .
Definition 4.21. For a -linear additive category , the Chern character is the -linear map
Proposition 4.22 (Proposition 2.4 in [5]). If is a perfect field and is Krull-Schmidt with a finite-dimensional endomorphism algebra for each indecomposable object, then the Chern character is injective.
Using these tools, we can now prove:
Theorem 4.23. Let . Then is infinite-dimensional in bidegree .
Proof. We let and again have isomorphisms
and we consider the category as a full subcategory of the enriched morphism category .
The -linear, additive category is Krull-Schmidt and hence idempotent complete; see e.g. the discussion in [30, Sections 4.5, 4.8] based on Bar-Natan’s category, which is equivalent to by [6].
Now may be considered as an additive, idempotent complete full subcategory of ; it is thus itself Krull–Schmidt. We have by Fact 4.12. Therefore, it suffices to compute its zeroth Hochschild homology of .
It is straightforward to check that the objects of have finite-dimensional endomorphism algebras, and since is perfect, the Chern character
is injective; see Proposition 4.22. To prove that is infinite-dimensional in bidegree , it is thus sufficient to show that is infinite-dimensional.
Moreover, is free abelian on the isomorphism classes of its indecomposable objects; cf. Proposition 4.20. Thus, we will be done once we can exhibit infinitely many indecomposable and pairwise non-isomorphic complexes appearing as (direct summands in) tangle complexes.
We will see that such complexes can be constructed as invariants of braids. Clearly, for there are infinitely many braids on strands. Moreover, the braid complexes are invertible under tensoring with the complex for the respective inverse braid. Since the complex of the trivial braid is indecomposable (its endomorphism algebra is local), so are the complexes for all other braids. It is also known that all braid complexes are pairwise non-isomorphic. This can e.g. be deduced from the faithfulness of the braid group action of Khovanov–Seidel [22]. For us, however, it is enough to consider infinitely many braids that are powers of a single Artin braid generator. For these complexes it is straightforward to check by hand that they are pairwise non-isomorphic. ∎
4.6. Comparison with the Rozansky–Willis invariant
In [31], Rozansky defined a Khovanov-type homology theory for (null-homologous) links in . His construction was generalized by Willis in [33] to null-homologous links in for any . We will denote the Rozansky-Willis homology of by . Just like the skein lasagna module , the invariant can be computed from a Kirby diagram for including the link , so it is a natural question whether they are related.
The first observation is that the two invariants are not always isomorphic. Indeed, in any specific bidegree, is defined as the Khovanov homology of the link in obtained from by adding sufficiently many twists in place of the 1-handles. It follows that has finite rank in each bidegree, whereas this may not hold for , as we have seen in Theorem 4.23. Another concrete example is for , where yields a -dimensional lasagna skein module according to Proposition 4.15, but is infinite-dimensional.
However, and are conceptually similar, as both arise as the Hochschild homology of a chain complex associated to a tangle that closes to the link :
- •
is computed as the Hochschild homology of a dg bimodule (for a tensor product of of Khovanov’s arc rings) associated to the tangle , as defined for by Khovanov in [18] and extended by parabolic induction to . Here the homological degree of the dg bimodule gets mixed with the Hochschild degree, and so the resulting invariant is a bigraded vector space.
- •
can be computed via Theorem 4.7 (and for even more concretely in Corollary 4.10) as the zeroth Hochschild homology of an equivalent dg bimodule; see Remark 4.13 for the comparison. In fact, the higher blob homology from [27], which does not play a role for skein lasagna modules, corresponds to higher Hochschild homology. The main difference, however, is that the dg bimodule is not considered as an object of a dg or triangulated category, but of the linear cohomology category. Accordingly, the full blob homology is triply-graded, with the blob/Hochschild grading separated from the homological grading.
Based on this comparison, one may expect and, more generally, the full blob homology to appear on the page of a spectral sequence converging to . Suppose that one can find a suitable projective resolution in terms of tangle complexes, which simultaneously allows the computation of blob homology as well as the dg version of Hochschild homology. Then, by tensoring with the dg bimodule associated to the tangle, one obtains a double complex of (quantum) graded vector spaces, where the vertical differential carries Hochschild degree and the horizontal differential carries homological degree. The homology of the total complex would compute . To obtain , one first takes homology in the rows (thus computing the Khovanov homologies of links of the form where appears in the resolution), and only then the zeroth homology of the induced differential coming from the resolution. We will not pursue this comparison further in the present paper, but remark that there is precedent for interesting invariants appearing on pages of spectral sequences that come from separating Hochschild and homological degrees, namely the triply-graded HOMFLYPT link homology; see [29, Section 6].
In general, one does not expect a map from the page of a spectral sequence to its page. However, since appears as the lowest row on the page, the above discussion suggests the existence of a natural map
In the following we propose a candidate for such a map.
In Willis’s construction of , we represent by pairs of spheres in the plane, with the spheres in each pair being identified (that is, we add a handle). This is the same as the usual Kirby diagram of . The link may intersect each handle a number of times, as in this picture:
Let be the link in obtained from by inserting full twists in place of the handle, as shown here:
The homology can be computed as the Khovanov homology of the link for , with some suitable shifts in grading. Note that depends on the choice of a path between the attaching spheres of each 1-handle; however, it can be shown that is independent of these choices up to isomorphism.
Consider now the skein lasagna module . Let us attach an -framed 2-handle through the 1-handle:
The 2-handles cancel the corresponding 1-handles, so the result is a Kirby diagram for , whose boundary is . The link becomes , as can be seen by doing a series of handle slides of the arcs of over the 2-handle:
where in the last step we cancelled the handles. (Compare Figure 5.13 in [10].)
The 2-handle attachments give a cobordism from to . There is also an embedded annular cobordism from to . As discussed in Section 2.2, these cobordisms induce a map on skein lasagna modules:
Our conjecture is that these maps stabilize as , giving a well-defined morphism from to .
4.7. Speculations on homotopy coherent four-manifold invariants
We expect that the above -page-of-spectral-sequence relationship between and for generalizes to for arbitrary four-manifolds and links . We give a brief sketch of the reasoning below.
Recall that the Khovanov-Rozansky invariants upon which is built assign chain complexes to links and chain maps to link cobordisms, but it is not known that this assignment is functorial (or even well-defined) at the level of complexes. The proof that the homology of these complexes is functorial in the appropriate sense involves showing that certain chain maps are homotopic. If this result could be strengthened to show that certain homotopies between the chain maps are themselves 2nd-order homotopic, and so on for all higher orders, then one could construct a functorial assignment of chain complexes to links in and chain maps to link cobordisms.
Let us assume that these conjectured “fully coherent” chain complexes for links exist. Then, they can be repackaged as a pivotal -category (with composition maps defined in terms of link cobordisms, as in [27]). This -category can in turn be fed into the machinery of Section 6.3 of [26] (which is closely related to topological chiral homology [23] and factorization homology [1, 2]). The result is a chain-complex-valued invariant . Its construction involves taking a homotopy colimit of a poset built out of the set of all ball decompositions of and refinement relationships between these ball decompositions. Concretely, we construct a double complex, with horizontal differentials coming from the complexes of links, and vertical differentials coming from the combinatorics of refining ball decompositions of . There is a spectral sequence associated to this double complex, which is itself an invariant of .
The page of this spectral sequence involves first taking homology in the horizontal direction, then computing homology with respect to vertical differentials. It is easy to see that this page is exactly the blob homology assigned to in [27] (i.e. by taking homology early instead of working with the complex). (In this paper we have focused on blob-degree zero, corresponding to the bottom row of the page of the spectral sequence.)
When and , we expect the total homology of to coincide with the Rozansky–Willis invariants. The Hochschild differentials of the previous subsection should be (homotopy equivalent to) special cases of the vertical differentials above.
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