HOMFLYPT homology for links in handlebodies via type A Soergel bimodules

We define a triply-graded invariant of links in a genus g handlebody, generalizing the colored HOMFLYPT (co)homology of links in the 3-ball. Our main tools are the description of these links in terms of a subgroup of the classical braid group, and a family of categorical actions built from complexes of (singular) Soergel bimodules.


Introduction
The HOMFLYPT polynomial is a classical invariant of links ⊂ 3 in the 3-sphere 3 with interesting and deep connections to representation theory. As pioneered by Jones [Jon87], the HOMFLYPT polynomial may be defined using representations of the classical n strand braid group ℬr(n) on the type A Hecke algebra. Indeed, we may use Alexander's theorem to present a link as a braid closure, and the HOMFLYPT polynomial then results by mapping the braid to the Hecke algebra and applying the so-called Jones-Ocneanu trace. (For the duration of the introduction, if not explicitly stated otherwise, "Hecke algebra" and related notions are always of type A.) This approach to the HOMFLYPT polynomial was categorified in work of Khovanov [Kho07]. In this work, Khovanov shows that the triply-graded Khovanov-Rozansky homology HHH • ( ) of ⊂ 3 , originally defined in [KR08b], admits a construction paralleling Jones's approach at the categorical level. This approach proceeds by replacing the Hecke algebra by the corresponding Hecke category, i.e. the category of Soergel bimodules. The latter admits a categorical action of ℬr(n) via so-called Rouquier complexes [Rou06], and the link homology results by taking Hochschild (co)homology, which provides a categorical analogue of the Jones-Ocneanu trace.
In addition to their triply-graded invariant, for each m ≥ 2 Khovanov and Rozansky define a doubly-graded homology theory for links ⊂ 3 [KR08a] that categorifies the sl m specialization of the HOMFLYPT polynomial. In the m = 2 case, which coincides with Khovanov's categorification of the Jones polynomial [Kho00], Asaeda-Przytycki-Sikora have extended this link homology to links in 3-manifolds ℳ = 3 [APS04], namely to links in thickened surfaces. Of particular interest is the case of the thickened annulus, where the so-called annular Khovanov homology has deep connections to both Floer theory and representation theory, see e.g. [Rob13], [GW10] and [GLW18]. In [QR18], an analogue of doubly-graded Khovanov-Rozansky homology was constructed for annular links, extending annular Khovanov homology, and its connection to representation theory, to general m. Unfortunately, the above approaches to link homology in 3-manifolds ℳ = 3 do not extend to the triply-graded setting.
In this paper, we remedy this by constructing generalizations of the triply-graded link homology for links in 3-manifolds distinct from the 3-sphere, namely in genus g handlebodies. (For g = 1, this is the case of links in the thickened annulus.) Our key insights are: (1) to consider various generalizations of the classical braid group that are related to links in handlebodies, and (2) that certain structures in categorical representation theory model the topology of the handlebody. We now detail our approach.
1A. An overview of our construction. Throughout, we let g, n ∈ N 0 . Recall that Khovanov's construction of HHH • ( ) for ⊂ 3 requires the following.
• Alexander's Theorem, which states that, up to isotopy, any link ⊂ 3 can be presented as the closure of a braid in the classical n-strand braid group ℬr(n).
• Markov's Theorem, which gives necessary and sufficient conditions for two distinct braids to have isotopic closures.
• A categorical action of ℬr(n) on the Hecke category via Rouquier complexes, which allows for the assignment of a chain complex of Soergel bimodules to each ∈ ℬr(n).
• Hochschild (co)homology, which produces a Markov invariant triply-graded vector space from this complex of Soergel bimodules.
In [HOL02] (see also [Lam93]), it is shown that analogues of Alexander's and Markov's Theorems hold for links in the genus g handlebody ℋ g . Playing the role of the classical braid group is the n-strand braid group ℬr(g, n) of the g-times punctured disk 2 g . The classical story here is the g = 0 case, where ℬr(n) = ℬr(0, n).
As we more fully detail in Section 2B, braids in ℬr(g, n) can be pictured as classical braids in the presence of non-intersecting "core strands". We then obtain a link in ℋ g by allowing the tops and bottoms of the core strands to meet at ∞, and by taking a closure of the "usual strands". The latter then form a link in the complement of the (glued) core strands, which is a handlebody ℋ g : = core strands usual strand (1-1) The analogue of the Alexander Theorem here shows that, up to isotopy, every link in ℋ g arises in this way, and the corresponding Markov Theorem characterizes when distinct braids give rise to isotopic closures. Issues arise, however, when attempting to carry out the last two steps in the construction of triply-graded link homology in this setting. Indeed, for general g, the groups ℬr(g, n) are not known to be Artin-Tits groups (see Section 1B for further discussion), so, to our knowledge, there are no associated Soergel bimodules and/or Rouquier complexes. Further, the Markov Theorem has a weaker notion of conjugation than in the classical case, e.g. we even though the indicated (boxed) braids , ∈ ℬr(2, 1) are conjugate. Hence, even with a categorical representation of ℬr(g, n) in hand, one cannot simply apply Hochschild cohomology to obtain an invariant of links that is sensitive to the topology of ℋ g .
We simultaneously resolve both these problems as follows. We expand the point at infinity to a small segment, which we move close to the top of the core strands. As a result, we can view the closure of the "usual strands" as a link in the handlebody given by the complement of the graph determined by the core strands and the segment at infinity, e.g.
In this modified presentation we are able to assign an invariant to the link ⊂ ℋ g using known structures in categorical representation theory. Indeed, for any labeling of the core strands, the boxed diagram in (1-3) determines a complex of singular Soergel bimodules. The latter determine a (2-)category that contains the Hecke category [Wil11], and categorifies the Schur algebroid, a certain idempotent completion of the Hecke algebra. Further, the closure procedure now does not involve the point at infinity, and hence can be carried out algebraically as usual, using Hochschild cohomology. In this way, we obtain a triply-graded homology for links ⊂ ℋ g . We show that this indeed produces a well-defined invariant of handlebody links, and that it is sensitive to the topology of the handlebody, e.g. it distinguishes the links in (1-2).
1B. A digression on Artin-Tits groups. Our motivation for this project, from which we have now somewhat strayed, was to further our understanding of the connection between low-dimensional topology and Artin-Tits groups. Recall that a Coxeter diagram Γ = (V, E) consists of a simple, complete graph with finitely many vertices V whose edges e = (i, j) ∈ E carry a label m ij = m ji ∈ N ≥2 ∪ {∞}. To any such diagram, we may associate the Artin-Tits group: (1-4) This group is an extension of the corresponding Coxeter group: The jumping-off point is the classical observation that ℬr(n) = ℬr(0, n) is isomorphic to the Artin-Tits braid group of type A, while ℬr(1, n) is isomorphic to the Artin-Tits group of type C = B and extended affine type A. More-surprising is the lesser-known fact that ℬr(2, n) is isomorphic to the Artin-Tits group of affine type C [All02]. The following table summarizes these known connections, details of which can be found in e.g. [All02, Section 4] and [Bri73]. ? (1-6) Herein, A n−1 denotes the type A Coxeter diagram with n − 1 nodes, whileÃ n−1 denotes the affine type A Coxeter diagram with n nodes and andÂ n−1 is the corresponding extended affine type. Similarly, C n andC n denote the type C = B and affine type C (but not affine type B) Coxeter diagrams with n and n + 1 nodes, respectively. As mentioned above, the first row of the type A column in (1-6) underpins Jones's construction of the HOMFLYPT polynomial, and the second row has similarly been exploited in topological considerations, see e.g. [OR07] and [El18]. The type C column, however, has received less attention, especially in the affine, g = 2 case, where not much appears to be known about connections to link invariants. (However, this case has been explored from a representation-theoretic point of view, see e.g. [DR18].) A notable example is work of Geck-Lambropoulou [GL97] in the g = 1 case, where a HOMFLYPT polynomial for links in ℋ 1 is constructed via the analogue of Jones's construction in type C. The results in [Rou17] and [WW11] should pair to give a categorification of this invariant. In a companion paper [RT], we plan to study this invariant, and develop its genus two analogue, using type C and affine type C Hecke algebras and Soergel bimodules.
By contrast, our construction in the present paper exploits the relation between ℬr(g, n) and ℬr(g + n), and the fact that the latter is an Artin-Tits group. Indeed, our construction can be recast as follows. By viewing the distinguished strands as usual strands, we obtain an injective group homomorphism ℬr(g, n) → ℬr(g + n). Since the latter is an Artin-Tits group, we can assign a complex of Soergel bimodules to any braid ∈ ℬr(g, n). Now, before taking Hochschild cohomology (doing so immediately would give an invariant less-sensitive to the topology of the handlebody), we glue on an additional Soergel bimodule that allows invariance under the Markov Theorem for ℬr(g, n), but not for ℬr(g + n). In fact, our procedure is slightly more general in that it uses an embedding of ℬr(g, n) into the colored braid group, and singular Soergel bimodules.
1C. Future outlook. In addition to our planned investigation in type C [RT], we believe there are a number of interesting future directions.
• The relation between ℬr(g, n) and Hecke algebras. These exists a Hecke-like algebra associated to ℬr(g, n) for general g, see e.g. [Lam00]. In the g = 0, 1 cases, this algebra matches the Hecke algebras associated to the Artin-Tits groups in the type A column of (1-6). These algebras have not been widely studied, e.g. to our knowledge it is not known whether they admit Markov traces or categorifications. In another direction, it is an interesting problem to extend the type C column of (1-6) to higher genus. The presentation of ℬr(g, n) given below in Definition 2.4 hints to a connection to the Artin-Tits group associated to the Coxeter diagram that is obtained from the type A n diagram by adjoining g additional vertices. These vertices are attached to each other with ∞-labeled edges, and to the first type A vertex with 4-labeled edges. (Something very similar was also observed in [Lam00, Remark 4].) For example, the g = 3 case is as follows: case g = 3 : Here, we depict k-labeled edges (for k < ∞) as k − 2 unlabeled edges. In fact, ℬr(g, n) is a quotient of the associated Artin-Tits group, so one could hope to extract invariants of ⊂ ℋ g from (a suitable quotient of) the corresponding Hecke algebra and/or Soergel bimodules.
• Connections to algebraic geometry. Work of Webster-Williamson [WW11] relates the Jones-Ocneanu trace on the type A Hecke algebra to the equivariant cohomology of sheaves on SL n , and extends this to other types. It would be interesting to identify geometry related to ℬr(g, n) and, more generally, links in ℋ g . One fertile avenue is the possible connection between the g = 2 case and exotic Springer fibers as e.g. in [SW18].
In a different direction, work of Gorsky-Negut-Rasmussen [GNR16] conjectures a relation between the category of type A Soergel bimodules and the flag Hilbert scheme of C 2 . The appearance of the latter can be interpreted as considering the closure of a braid ∈ ℬr(n) in the complement of an n-component unlink. Since the graph giving the complement of ℋ g can be viewed as an unlink fused at the "segment at infinity," this suggests a connection between the flag Hilbert scheme and our invariants.
1D. Conventions. We now summarize various conventions used in this paper.
Convention 1.1. We work over an arbitrary field K of characteristic 0. This requirement is only needed in Section 4: the reader interested in integral versions of our results from Section 3 needs to replace the algebraic definition of singular Soergel bimodules of type A, which we use, by their diagrammatic incarnation [EL17, Section 2.5]. (The algebraic and the diagrammatic definitions differ when working integrally or in characteristic p.) All the results from Section 3 then hold verbatim over Z. However, we do not currently have integral versions of the singular Soergel diagrammatics in Section 4.
Convention 1.2. We will find it convenient to depict morphisms in certain categories (and 1-morphisms in certain 2-categories) diagrammatically. We will read such diagrams from bottom-to-top (and in the presence of a monoidal or 2-categorical structure, also right-to-left). These reading conventions are summarized by Moreover, all such diagrams are invariant under (distant) height exchange isotopy (up to isomorphism, in the 2-categorical context). Finally, we will occasionally omit certain data (e.g. labelings) from such diagrams when it may be recovered from the given data, or is not important for the argument in question.
Convention 1.3. We will work with Z k -graded categories throughout, for k = 1, 2, 3. The three gradings of importance are the internal degree q, the homological degree t (both appearing from Section 3 onward), and the Hochschild degree a (making its appearance in Section 4).
There are competing notions of what is meant by a graded category, so we now detail our conventions, focusing on the q-degree. Let C be a category enriched in Z-graded abelian groups, i.e. for objects X and Y, Hom C (X, Y) is a Z-graded abelian group: Given such a category, we can introduce a formal grading-shift functor q and consider the category C q in which objects are given by formal shifts q s X of objects in C, and (1-10) i.e. C q is again enriched in Z-graded abelian groups. Finally, we let C q be the category with the same objects as C q , but where we restrict to q-degree zero morphisms, i.e.
(1-11) Note that C q is not enriched in Z-graded abelian groups, but is equipped with an autoequivalence shift functor q. It is categories of this form that will be of primary interest in this work.
We note, however, that it is possible to recover the Z-graded abelian group Hom C (X, Y) from the category C q . Indeed, we can consider the Z-graded abelian group and we note that where on the right-hand side the power q denotes a shift of the indicated Z-graded abelian group.
Our consideration of Z 2 -and Z 3 -graded categories is analogous -in these cases we have additional shift functors t and a, and we restrict to t-and a-degree zero maps, unless otherwise indicated. However, we will reserve the capitalization notation HOM when considering "graded Homs" with respect to the q-degree only.
Lastly, we note that these considerations carry over to 2-categories as well, where the above applies to the Hom-categories in our 2-category, i.e. to the 1-and 2-morphisms. Universität Zürich, and he thanks them for excellent working conditions. It continued during visits of both authors to the Kavli Institute for Theoretical Physics for the program Quantum Knot Invariants and Supersymmetric Gauge Theories. As such, this research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. We thank them for the engaging and collaborative atmosphere. Some of the ideas for this project came up during a discussion of the second author and Catharina Stroppel, and he thanks her for freely sharing her ideas about HOMFLYPT homology outside of type A.
Moreover, D.E.V.R. thanks Andrea Appel, Ben Elias, and Matt Hogancamp for useful discussions, and D.T. thanks Ben Elias, Aaron Lauda, Anthony Licata, Catharina Stroppel, Emmanuel Wagner and Arik Wilbert for related discussions about the topology behind Artin-Tits groups, Mikhail Khovanov, Paul Wedrich and Oded Yacobi for helpful conversations and encouraging words, and also his office chair for supporting this project.
Finally, D.E.V.R. was partially supported by Simons Collaboration Grant 523992, and D.T. is grateful to NCCR SwissMAP for generous support that, in particular, partially financed the first author's visit to Zürich.

Links and braids in handlebodies
In this section we collect result concerning links and braids in handlebodies.
2A. Topological recollections. Recall that a handlebody ℋ g of genus g is the compact, orientable 3-manifold with boundary obtained by attaching g 1-handles to the closed 3-ball ℋ 0 . An explicit model for ℋ g , that we call the standard presentation, is given by the "inside" of a standardly embedded genus g surface Σ g ⊂ 3 in the 3-sphere 3 , i.e. the 3-manifold given by the union of Σ g with the component of 3 Σ g that does not contain the point at infinity.
We will typically work with another presentation for ℋ g , given by the closure in 3 of the complement of an auxiliary handlebody ℋ c g . We view ℋ c g as consisting of g parallel 1-handles that are attached to the closure of a small neighborhood of ∞ ∈ 3 . See the gray portion of the first figure in Example 2.2 for the g = 3 case. In order to connect with the categorical representation theory used to produce our link invariant, we note that ℋ c g is isotopic to a closed neighborhood of the embedded graph obtained by taking g parallel edges, called the "core strands", each meeting a (g + 1)-valent vertex, together with an additional "edge at infinity" joining the two vertices. We will typically view the edge at infinity as being near the top of the core strands, hereby viewing ℋ c g as being obtained from the core strands by first gluing on a graph with g 1-valent vertices at its top and bottom (and two (g + 1)-valent vertices) and then taking their closure. See the gray portion of the second figure in Example 2.2.
We will refer to this presentation of ℋ g = 3 ℋ c g as the costandard presentation, and note that it contains a copy of the standard presentation, to which it is isotopic, given by intersecting with a closed 3-ball that meets each core strand in a segment.
We consider oriented links ⊂ ℋ g , which, in the costandard presentation, are given by links in 3 ℋ c g . Equivalently, using the isotopy with the standard presentation, such links are given by links in the closed 3-ball that avoid its intersection with the core strands. Finally, two links in , ⊂ ℋ g are isotopic, denoted by ∼ , if and only if the corresponding links in 3 (in the costandard presentation) are isotopic through an isotopy that keeps ℋ c g fixed pointwise.
2B. Alexander's theorem. There is a corresponding notion of n strand braids in a genus g handlebody. Strictly speaking, we define ℬr(g, n) to be the braid group of the surface 2 g given as the complement of g disjoint open disks in the closed disk 2 . The standard presentation of ℋ g may be identified with the product 2 g × [0, 1] ⊂ 3 , so braids in ℬr(g, n) "live in" ℋ g . The group ℬr(g, n), which we call the handlebody braid group, can be equivalently described as follows. There is a subgroup of the classical braid group ℬr(g + n) on g + n strands consisting of braids that are pure on the first g strands, and a homomorphism from this subgroup to the classical braid group ℬr(g) given by forgetting the final n strands. The kernel of this homomorphism is precisely ℬr(g, n). (Informally, ℬr(g, n) consists of braids on g + n strands in which the first g strands do not braid among themselves.) Slightly abusing notation, we will again refer to the first g strands of a braid in ℬr(g, n) as core strands; they correspond to the core strands above as we now describe.
The handlebody braid group ℬr(g, n) is related to links in ℋ g in a manner paralleling the relation between the classical braid group ℬr(n) and links in 3 . That is, given a braid ∈ ℬr(g, n), one obtains a link ⊂ ℋ g via a closure procedure as follows: the first g strands in are joined at each of their ends to the point at infinity, and the remainder of the braid is closed as in the classical case. In this way, we obtain a link ⊂ ℋ g where the closure of the last n strands constitutes , and the first g strands in become the core strands in ℋ c g . As in our discussion of ℋ c g above, we will typically work with an equivalent closure procedure, which again corresponds to expanding the point at infinity to an edge, and moving it near the top of the core strands. Specifically, the closure procedure consists of merging the g core strands to meet the strand at infinity, then splitting the strand at infinity into g strands, and finally taking the standard closure of all strands.
Example 2.1. We have ℬr(0, n) ∼ = ℬr(n), which corresponds to links in the closed 3-ball ℋ 0 ; we call this the classical case. In genus one, ℬr(1, n) consists of all braids in ℬr(1 + n) that are pure on the first strand, and ℋ 1 is a solid torus.
The braid itself is depicted as the solid strands in the indicated rectangle, while the dashed edges correspond to the closure procedure described above. The thin, black components (both solid and dashed) give the link , while the thick, gray graph (again, both solid and dashed) depicts ℋ c g .
The next result shows that, up to isotopy, all links in ℋ g arise from the closure procedure for handlebody braids described above. The proof is analogous to the classical case. 2C. Generators and relations for braids in handlebodies. We now recall the algebraic presentation of ℬr(g, n).
Definition 2.4. The group Br(g, n) is the group generated by 1 , . . . , n−1 and 1 , . . . , g , called braid and twist generators, respectively, subject to the relations By convention, Br(g, 0) = {1}, and we omit the twist generators when g = 0 and the braid generators when n = 1.
An explicit isomorphism realizing Proposition 2.5 is given on the braid and the twist generators as follows: The inverse of i is given, as usual, by the corresponding opposite crossing, and the inverse of i is given as above, but with the braid strand wrapping the i th core strand oppositely (however, it still crosses over the other core strands).
Remark 2.8. The conjugation (2-9) is weaker than in the classical case -there one can conjugate by any element, instead of just by certain elements. This will play an important role in our construction of the HOMFLYPT invariant, see Proposition 4.8. On the other hand, the stabilization (2-10) is stronger than the classical case when considered on its own, but together with the classical conjugation relation is equivalent to the classical stabilization. Remark 2.10. Although it may appear that Definition 2.7 omits conjugation by certain elements that clearly give isotopic closures e.g. by the "maximal loop" ω = g . . . 1 or its inverse, [HOL02, Section 5] shows how conjugation by such elements can be described in terms of the above Markov moves.
2E. From handlebody braids to classical braids. Recall from Section 1B that one of our main ingredients in constructing homological invariants of links in ℋ g for all g ≥ 0 is the relation between ℬr(g, n) and (a colored variant of) the type A braid group ℬr(g + n): We have a group homomorphism ℬr(g, n) → ℬr(0, g + n) given by viewing the core strands as "usual" strands, e.g. (2-11) As discussed above, this map is clearly injective, hence we have: Proposition 2.11. The map induced by (2-11) gives rise to an embedding of groups (2-12) ℬr(g, n) → ℬr(g + n).
However, Proposition 2.11 is only one ingredient in our construction, since invariance under the procedure (2-13) (i.e. under "conjugation in ℬr(g + n)") is not desirable for an invariant of ∼ ⊂ ℋ g , cf.
Remark 2.8. As such, we will use the theory of singular Soergel bimodules to mimic the merging and splitting of the core strands in the closure procedure for ℬr(g, n), which will lead to invariants of that are not invariant under (2-13).

Braids in handlebodies and singular type A Soergel bimodules
In the present section, we construct a map from ℬr(g, n) to the 2-category of singular Soergel bimodules.
x j else.
(3-1) By contrast, the representation given by (3-1) is built from the N -dimensional permutation representation, which decomposes as a direct sum of the standard representation and the trivial representation. By e.g. [EK10, Section 4.6] and [EL17, Theorem 2.7 and Proposition 2.10], the difference (akin to the difference between considering gl N rather than sl N ) will not play a role in the present work, in the sense that all results from the cited literature hold in this case as well.
Fix any tuple I = (k 1 , . . . , k r ) ∈ N r ≥1 with k 1 + · · · + k r = N . (Note further that choosing I also determines N since N = k 1 + · · · + k r . We will tacitly use this throughout.) By definition, the corresponding parabolic subgroup is Since there is a bijection between tuples and parabolic subgroups, we will implicitly identify them, e.g. I ⊂ J denotes an inclusion of parabolic subgroups.
For the duration, we will use the following ordering convention for parabolic subgroups I, J, K, L and their rings of invariants: The q-degree 0 inclusion ι J I : R J → R I (3-5) of K-algebras is a q-graded Frobenius extension (see [ESW17]), meaning that R I is a q-graded, free R J -module of finite rank, possessing a non-degenerate, R J -linear trace map R I → R J . In the present case, the latter is built using the Demazure operators The collection {∂ σ i } satisfies the classical braid relations, and thus gives a well-defined map ∂ w associated to any w ∈ S(N ) using a reduced expression for w. Using these, the aforementioned trace map is given by and is of q-degree 2 (I) − 2 (J). Here w I is the longest element in S I (N ), and (I) denotes its length.
The Frobenius extension data allows for the definition of maps between certain R I -bimodules, that will serve as important morphisms between singular Soergel bimodules (we recall the definition of the latter below). To wit, given a basis {a i } for R I over R J , we can find a dual basis {a i } satisfying ∂ J I (a i a j ) = δ ij . Given this, we obtain the Frobenius element a := i a i ⊗ a i , which is of q-degree 2 (J) − 2 (I) and independent of the choice of {a i }. This gives multiplication and comultiplication maps . (3-7) These morphisms of bimodules are unital and counital with respect to ι J I and ∂ J I , respectively. Example 3.3. For I = ∅ and J = {i}, we have w I = 1 and w J = σ i . It follows that {1, 1 2 α i } and {1 = 1 2 α i , ( 1 2 α i ) = 1} are dual bases for R as an R {i} -module, and a = 1 2 (1 ⊗ α i + α i ⊗ 1). Finally, for later use, let us explicitly identify the rings R I for all I = (k 1 , . . . , k r ). To this end, we consider r alphabets X i (we tend to omit the alphabets if no confusion can arise) with k i variables, and write ⊗ K = ⊗. A classical result about symmetric functions gives that R I ∼ = K e 1 (X 1 ), . . . , e k 1 (X 1 ) ⊗ · · · ⊗ K e 1 (X r ), . . . , e kr (X r ) , where e j (X i ) denotes the j th elementary symmetric function in the variables X i . Note that qdeg(e j ) = 2j. In particular, 3B. A reminder on type A singular Soergel bimodules. We now briefly recall the category of singular Soergel bimodules SS q (N ) = SS q (A N −1 ) of type A N −1 , which categorifies the Hecke/Schur algebroid of type A [Wil11, Theorem 1.2] in characteristic 0. Details (in more generality) can be found e.g. in [Wil11], or [EL17] and [ESW17] for the underlying diagrammatic calculus.
Define the merge ("restriction") and split ("induction") bimodules as follows: where we follow the conventions from (3-4). Here, we have indicated the left/right actions using left/right subscripts, a convention that we will use throughout. There is a (horizontal) composition of such bimodules given by tensoring over the common ("middle") ring, which we denote e.g. by L M J M I = L M J ⊗ R J J M I . In particular, we have the following q-degree 0 bimodule isomorphisms that we implicitly use below: All of the isomorphism in (3-11) are essentially identities, as the careful reader is invited to check. (Note e.g. for I ⊂ J, and 2-morphisms are (all) bimodule maps of q-degree 0.
Example 3.5. We have q -1 R ⊗ R {i} R ∼ = 1,1 S 2 M 1,1 , which the reader familiar with (usual) Soergel bimodules of type A (see e.g. [EW14]) might recognize as being so-called Bott-Samelson bimodules. In particular, Soergel bimodules of type A N −1 can be identified with the 2-category End SS q (N ) (∅), which has just one object (hence is a monoidal category).
The basic building blocks are the identity, merge, and split bimodules, which are depicted using the following (local) graphical notation: Here, and for the duration, we use the abbreviation R k for the ring associated to I = (k).
Recall that, by Convention 1.2, vertical concatenation of such pictures corresponds to composition of 1-morphisms, and e.g. composing on the left corresponds to stacking on the top. Moreover, we can place such diagrams side-by-side, which corresponds to taking the tensor product over K. Hence, we can associate a singular Soergel bimodule to each trivalent graph that we can build from these diagrams via these operations. (Note that webs corresponding to singular Soergel bimodules never have edges of negative label, but we will allow them in formulas for convenience of notation, with the understanding that the corresponding bimodules are zero.) Example 3.6. For N = 2, the standard way to depict the Soergel bimodule from Example 3.5 (see e.g. [Kho07, Figure 2]) is built into our conventions: Also of importance will be the ladder-rung bimodules:   Definition 3.8. Given an additive category C, we denote its bounded homotopy category by K b C . This is the category whose objects are bounded chain complexes, and whose morphisms are homotopy classes of chain maps. We will use to denote isomorphisms in K b C , i.e. homotopy equivalence.
Recalling Section 1D, we can view the objects in K b C as finite direct sums i t k i X i , equipped with a differential d with tdeg(d) = −1. There is a t-degree zero inclusion of categories C → K b C given by considering objects of C as one-term complexes concentrated in t-degree 0. We also remark that we can consider K b for a 2-category , by passing to the homotopy category in each Hom-category. In particular, if C is monoidal, then so is We now recall Rickard-Rouquier complexes, i.e. complexes of singular Soergel bimodules that determine maps from the (colored) braid group(oid) into certain Hom-categories in K b SS q (N ) . Our terminology here arises as these complexes correspond to the Rickard complexes (originally defined for symmetric groups) in categorified quantum groups, but also agree with the type A Rouquier complexes in the "uncolored" k = l = 1 case.
They are given as follows: where m = min(k, l). Our notation denotes e.g. that, as a tq-graded bimodule, β i k,l is the direct sum of the indicated terms, and the arrows depict the non-zero components of the differentials. Recalling the bimodule maps from (3-11), (3-5), (3-6), (3-7), and omitting the tq-shifts, these are given by Here the corresponding parabolic subsets, which determine the bimodule maps, can be read from the indicated sequence of webs, and we use e.g. ι J I as as well as variants with analogous q-shifts involving split bimodules.
Let I be a parabolic subgroup, and let #I denote the number of entries in the corresponding tuple (i.e. for an r-tuple I, #I = r). Given a braid generator i ∈ ℬr(#I), we let ±1 i I denote the complex given by placing appropriately labeled vertical strands next to the corresponding complex in (3-18), i.e. by taking tensor product over K with the rings R (k 1 ,...,k i−1 ) and R (k i+2 ,...,k #I ) .
Definition 3.13. For I and ∈ ℬr(#I), fix an expression = ±1 i 1 . . . ±1 ir . Define where, on the right-hand side, we use composition in K b SS q (N ) , i.e. tensor product of the complexes of singular Soergel bimodules.
By e.g. the results in [QRS18, Section 5.2], the complex I does not depend, up to isomorphism, on the choice of expression for . Thus, the assignment → I gives an action of ℬr(#I) on K b SS q (N ) . We get: Proposition 3.14. There is an action of ℬr(g, n) on K b SS q (N ) determined by the assignment → I . Proof. By the discussion above, we have an action of the classical braid group. Composing this action with the map from Proposition 2.11 gives the desired action of the handlebody braid group.

Colored HOMFLYPT homology for links in handlebodies
In this section, we proceed to construct our triply-graded invariant of links in ℋ g , with Theorem 4.7 and Corollary 4.13 being the main statements. We keep the notation from the previous sections and begin with some preliminaries. 4A. A reminder on Hochschild cohomology. Let A be a q-graded K-algebra, and recall that we may regard any q-graded A-bimodule B as a q-graded left module over the enveloping algebra A ⊗ A op . The Hochschild cohomology of A with coefficients in B is the aq-graded K-vector space The relevant case for our considerations is when A = R I = (R I ) op . Here, for I = ∅, Khovanov showed that the triply-graded link homology from [KR08b] can be constructed using the Hochschild homology (defined using Tor instead of Ext) of Soergel bimodules; see [Kho07, Section 1.1]. Recall from (3-8) that R I is a polynomial ring, so Hochschild homology and cohomology are isomorphic (up to a shift). We work with the latter since e.g. in this framework the invariant of the (colored) unknot inherits a natural algebra structure [Hog18], which is important for various considerations.
Example 4.1. Let I = (k 1 , . . . , k r ). Recall that R I is a polynomial ring (and, in particular, is Koszul). Hence, we can compute Hochschild cohomology using the Koszul resolution of R I , which is the free resolution of R I as an R I -bimodule given by Here h denotes a shift up in an auxiliary homological degree, and the outer tensor products are taken over R I ⊗ R I . Given a R I -bimodule B, taking the "internal" q-graded Hom of complexes HOM • R I ⊗R I (−, B) (i.e. applying HOM R I ⊗R I (−, B) to the terms and differentials of a chain complex to obtain a cochain complex) gives a complex concentrated in non-negative cohomological degree a, which is the negative of the h-degree. The a th cohomology of this complex is HH a (R I , B).
Computing for B = R I gives the following. For each j, fix a set of variables −2i), and recall that aqdeg(e i ) = (0, 2i). We then have an isomorphism of aq-graded K-vector spaces Since Hochschild cohomology is functorial with respect to bimodule morphisms, we can apply HH • to a complex of R I -bimodules term-wise to obtain a complex of aq-graded Kvector spaces. (In fact, since our ring is commutative, these K-vectors spaces inherit an action of R I , so can be thought of as trivial R I -bimodules.) In particular, let R I Bim q denote the category of q-graded, finitely-generated R I -bimodules, and let K b R I Bim q be its homotopy category. We get a functor 4B. Towards handlebody HOMFLYPT homology. Fix integers M, l 1 , . . . , l n ∈ N ≥1 , called the core and link colors, respectively. For any g ≥ 0, these choices determine a parabolic subset M := (M, . . . , M, l 1 , . . . , l n ) with #M = g + n. We view M as providing a coloring for braids ∈ ℬr(g, n) as in Section 3D, where strands are colored at the bottom by the entries of M. We will call a colored braid ( , M) balanced if the colors at the top and bottom of the i th position agree for all i. For the duration, we only consider balanced colorings and any braid or link will be colored by default.
Example 4.2. The prototypical example of a balanced coloring is the case where the link is uncolored, i.e. where l 1 = · · · = l n = 1 and M is arbitrary. In general, M should be viewed as being "very large," i.e. M l i for all i; compare e.g. to [ILZ18], where the core of the solid torus is colored by a Verma module.
Remark 4.3. It is possible to work with any balanced coloring of ∈ ℬr(g, n). However, the core strands are not topologically distinguishable, hence should be colored uniformly.
Consider SS q (I) := End SS q (N ) (I) which is a q-graded, full, monoidal subcategory of R I Bim q . The monoidal structure is inherited from the horizontal composition in SS q (N ), i.e. it is given by tensor product over R I . We will occasionally denote this by ⊗ R I , in addition to our previous notation for this operation, which was simply concatenation.
The remainder of this section constitutes a proof of this theorem. Namely, invariance under conjugation holds as a special case of the corresponding result for colored, triply-graded link homology in 3 , and Lemma 4.12 establishes invariance (up to a grading shift) under stabilization. Proposition 4.8 shows the failure of invariance under the classical conjugation relation. We stress the importance of this latter fact: any invariant of the classical braid group ℬr(g + n), that additionally is invariant under classical conjugation and stabilization, gives rise to invariants of links in ℋ g using the inclusion ℬr(g, n) → ℬr(g + n). However, such invariants are less-sensitive to the topology of ℋ g , as our results show.
Proposition 4.8. For g > 1 and n > 0, there exists handlebody braids , ∈ ℬr(g, n) that are conjugate in ℬr(g + n), but satisfy HH • ℋg ( ) HH • ℋg ( ). This shows that our handlebody homology, which is defined below to be the cohomology of a renormalization of HH • ℋg (−), distinguishes these handlebody links, while the invariant obtained by including into ℬr(g + n) and using the classical (colored) triply-graded homology does not.
Proof. It suffices to give an example, and we provide one in the g = 2 and n = 1 case that immediately generalizes to any g ≥ 2 and n ≥ 1. Let = 2 1 and = 1 2 , which are conjugate braids in ℬr(g + n). We claim that HH • ℋg ( ) HH • ℋg ( ), and exhibit this explicitly in the case that M = 1.
Indeed, if they were homotopy equivalent, then the Euler characteristics (i.e. alternating sums of aq-graded dimensions) of these complexes would agree. However, since the category of (usual) type A Soergel bimodules categorifies the type A Hecke algebra, and Hochschild cohomology categorifies the Jones-Ocneanu trace, this would imply that the Jones-Ocneanu traces of the following braided, trivalent graphs agree: Using the decategorification of the first equation in Example 3.10, this in turn would imply that the HOMFLYPT polynomials of the links given as the closures of & (4-12) agree. However, a computation shows that the difference between their (reduced) HOMFLYPT polynomials is (a − a -1 ) 2 − (q − q -1 ) 2 , where a, q are variables (at the decategorified level) corresponding to a, q.
We now turn out attention to the behavior of HH • ℋg ( ) under stabilization (2-10). Our main technical tool will be the partial Hochschild trace from [Hog18, Section 3], which we now adapt to the colored setting. The construction of this functor is motivated as follows. Since Hochschild cohomology satisfies the classical conjugation relation (i.e. the relation (2-13)), we informally view this operation as a mean to take the closure of (the singular Soergel bimodule associated to) a web diagram. In order to study the stabilization relation, we would like to be able to take this closure "one strand at a time" in a manner that is compatible with taking Hochschild cohomology.
Recall that the Hochschild cohomology of an R I -bimodule M is defined as where here we follow Convention 1.3 for the q-graded Hom. Here D b R I Bim q is the bounded derived category, and we emphasize that the homological degree therein is not the t-degree from Section 3D, but rather the h-degree from Example 4.1 (which should be viewed as "perpendicular" to the homological degree of the Rickard-Rouquier complexes). Our discussion above suggests that we should consider functors between the categories D b R I Bim q for various I that are compatible with the functors HOM D b (R I Bim q ) (R I , −).
To this end, given I = (k 1 , . . . , k r ) we let I − := (k 1 , . . . , k r−1 ), i.e. I − is obtained from I by removing the last entry. Let Q kr I := R I ⊗ R I (e i (X r ) ⊗ 1 − 1 ⊗ e i X r ) kr i=1 . Using the notation in (3-8), we have which suggests that we consider the functor I I : D b R I − Bim q → D b R I Bim q given by derived tensor product ⊗ L with Q kr I over R I − ⊗ R I − . We then obtain T I : D b R I Bim q → D b R I − Bim q , which we define to be the right adjoint to I I , using derived tensor-hom adjunction.
The functors T I and I I admit the following explicit descriptions. We have an isomorphism where the (outer) tensor product is taken over R I ⊗ R I . Since K kr I is a complex of free R I ⊗ R I -modules, given any complex M ∈ D b R I Bim q , T I (M) is the complex (In the case that M is a complex, we interpret the latter as a shift of the cone of the indicated chain map.) Similarly, for N ∈ D b R I − Bim q , we have that where we again interpret the latter as the total complex of this double complex. Here, since K kr I ∼ = Q kr I and the latter is a free R I − -bimodule, we also have  Additionally, given N, P ∈ D b R I − Bim q , we have Finally, setting I 1 := {k 1 , . . . , k s } and I 2 := {k s+1 , . . . , k r }, we have for M 1 ∈ D b R I 1 Bim q and M 2 ∈ D b R I 2 Bim q .
We will use the functors I I and T I to give a "local" proof of invariance under stabilization. Note that there is a q-degree 0, fully faithful inclusion functor SS q (I) → D b R I Bim q given by viewing a singular Soergel bimodule as a complex concentrated in h-degree zero. Further, this functor is monoidal (with respect to ⊗ R I on the former and ⊗ L R I on the latter) since singular Soergel bimodules are free as either left or right R I -modules. (The latter can be deduced from the fact that R I is free over R J for I ⊆ J, cf. Section 3A.) Given this, we now develop a graphical interpretation for the action of the functors I I and T I on singular Bott-Samelson bimodules, again adapting [Hog18, Section 3.3] to the singular setting. Since our eventual aim is to apply these results to HH • ℋg (−), we will focus on the (bimodules appearing in the) complex Similarly, taking Hochschild cohomology will be depicted by closing all (non-core and core) strands. In this language, the first statement in Lemma 4.9 says that we obtain the same result whether we close all strands at once or one at a time, while the second and third are Next, we compute the value of the colored partial trace on the "merge-split" bimodule. (Strictly speaking, we will only use the k = l = 1 case of Lemma 4.10, which is given e.g. in [Hog18,Equation (3.1b)]. However, as we are developing the skein calculus for the colored partial trace, and since we anticipate applications of this formula to explicit computations of our invariant, we take the opportunity to extend loc. cit. to the colored setting.) Proof. In the k = 0 case, the result simply claims that the l-colored circle is a K-vector space of aq-graded dimension l i=1 1+aq -2i 1−q 2i . This follows directly from Example 4.1. We thus assume that k ≥ 1, and proceed as in the proof of [Hog18, Proposition 3.10]. Namely, we explicitly write down the value of T I on the bimodule in the left-hand side of (4-24), apply a change of variables, and use this to explicitly identify the result in the derived category.
Finally, the second statement follows from (the proof of) Proposition 4.8, since this shows that the homology of HH • ℋg for the braids therein are not isomorphic up to a degree shift.