HOMFLYPT homology for links in handlebodies via type A Soergel bimodules
HHOMFLYPT HOMOLOGY FOR LINKS IN HANDLEBODIES VIA TYPE A SOERGEL BIMODULES
DAVID E.V. ROSE AND DANIEL TUBBENHAUER
Abstract.
We define a triply-graded invariant of links in a genus g handlebody, generalizingthe colored HOMFLYPT (co)homology of links in the -ball. Our main tools are thedescription of these links in terms of a subgroup of the classical braid group, and a familyof categorical actions built from complexes of (singular) Soergel bimodules. Contents
1. Introduction 12. Links and braids in handlebodies 73. Braids in handlebodies and singular type A Soergel bimodules 104. Colored HOMFLYPT homology for links in handlebodies 15References 241.
Introduction
The HOMFLYPT polynomial is a classical invariant of links 𝓁 ⊂ 𝒮 in the -sphere 𝒮 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 strandbraid group ℬ r( n ) on the type A Hecke algebra. Indeed, we may use Alexander’s theorem topresent a link as a braid closure, and the HOMFLYPT polynomial then results by mappingthe braid to the Hecke algebra and applying the so-called Jones–Ocneanu trace. (For theduration of the introduction, if not explicitly stated otherwise, “Hecke algebra” and relatednotions 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 • ( 𝓁 ) of 𝓁 ⊂ 𝒮 , originally defined in [KR08b], admits a construction paralleling Jones’sapproach at the categorical level. This approach proceeds by replacing the Hecke algebraby the corresponding Hecke category, i.e. the category of Soergel bimodules. The latteradmits a categorical action of ℬ r( n ) via so-called Rouquier complexes [Rou06], and the linkhomology results by taking Hochschild (co)homology, which provides a categorical analogueof the Jones–Ocneanu trace.In addition to their triply-graded invariant, for each m ≥ Khovanov and Rozanskydefine a doubly-graded homology theory for links 𝓁 ⊂ 𝒮 [KR08a] that categorifies the sl m specialization of the HOMFLYPT polynomial. In the m = 2 case, which coincides withKhovanov’s categorification of the Jones polynomial [Kho00], Asaeda–Przytycki–Sikora haveextended this link homology to links in -manifolds ℳ (cid:54) = 𝒮 [APS04], namely to linksin thickened surfaces. Of particular interest is the case of the thickened annulus, wherethe so-called annular Khovanov homology has deep connections to both Floer theory andrepresentation theory, see e.g. [Rob13], [GW10] and [GLW18]. In [QR18], an analogue ofdoubly-graded Khovanov–Rozansky homology was constructed for annular links, extending a r X i v : . [ m a t h . QA ] A ug DAVID E.V. ROSE AND DANIEL TUBBENHAUER annular Khovanov homology, and its connection to representation theory, to general m .Unfortunately, the above approaches to link homology in -manifolds ℳ (cid:54) = 𝒮 do not extendto the triply-graded setting.In this paper, we remedy this by constructing generalizations of the triply-graded linkhomology for links in -manifolds distinct from the -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 inhandlebodies, and (2) that certain structures in categorical representation theory model thetopology of the handlebody. We now detail our approach.1A. An overview of our construction.
Throughout, we let g, n ∈ N . Recall thatKhovanov’s construction of HHH • ( 𝓁 ) for 𝓁 ⊂ 𝒮 requires the following. • Alexander’s Theorem, which states that, up to isotopy, any link 𝓁 ⊂ 𝒮 can bepresented as the closure of a braid 𝒷 in the classical n -strand braid group ℬ r( n ) . • Markov’s Theorem, which gives necessary and sufficient conditions for two distinctbraids to have isotopic closures. • A categorical action of ℬ r( n ) on the Hecke category via Rouquier complexes, whichallows for the assignment of a chain complex of Soergel bimodules to each 𝒷 ∈ ℬ r( n ) . • Hochschild (co)homology, which produces a Markov invariant triply-graded vectorspace from this complex of Soergel bimodules.In [HOL02] (see also [Lam93]), it is shown that analogues of Alexander’s and Markov’sTheorems hold for links in the genus g handlebody ℋ g . Playing the role of the classical braidgroup is the n -strand braid group ℬ r( g, n ) of the g -times punctured disk 𝒟 g . The classicalstory here is the g = 0 case, where ℬ r( n ) = ℬ r(0 , n ) .As we more fully detail in Section 2B, braids in ℬ r( g, n ) can be pictured as classical braidsin the presence of non-intersecting “core strands”. We then obtain a link in ℋ g by allowingthe tops and bottoms of the core strands to meet at ∞ , and by taking a closure of the “usualstrands”. The latter then form a link in the complement of the (glued) core strands, which isa handlebody ℋ g : 𝒷 = core strandsusual strand ⊂ 𝒟 g × [0 , closure −−−−→ • ∞ • ∞ (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 braidsgive rise to isotopic closures.Issues arise, however, when attempting to carry out the last two steps in the constructionof triply-graded link homology in this setting. Indeed, for general g , the groups ℬ r( g, n ) are not known to be Artin–Tits groups (see Section 1B for further discussion), so, to ourknowledge, 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 OMFLYPT HOMOLOGY FOR LINKS IN HANDLEBODIES 3 have • ∞ • ∞ 𝒷 not isotopic ←−−−−−−→ • ∞ • ∞ 𝒸 (1-2)even though the indicated (boxed) braids 𝒷 , 𝒸 ∈ ℬ r(2 , are conjugate. Hence, even with acategorical representation of ℬ r( g, n ) in hand, one cannot simply apply Hochschild cohomologyto 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 infinityto a small segment, which we move close to the top of the core strands. As a result, we canview the closure of the “usual strands” as a link in the handlebody given by the complementof the graph determined by the core strands and the segment at infinity, e.g. • ∞ • ∞ ∼ (1-3)In this modified presentation we are able to assign an invariant to the link 𝓁 ⊂ ℋ g usingknown structures in categorical representation theory. Indeed, for any labeling of the corestrands, the boxed diagram in (1-3) determines a complex of singular Soergel bimodules. Thelatter determine a ( -)category that contains the Hecke category [Wil11], and categorifies theSchur algebroid, a certain idempotent completion of the Hecke algebra. Further, the closureprocedure now does not involve the point at infinity, and hence can be carried out algebraicallyas usual, using Hochschild cohomology. In this way, we obtain a triply-graded homology forlinks 𝓁 ⊂ ℋ g . We show that this indeed produces a well-defined invariant of handlebodylinks, and that it is sensitive to the topology of the handlebody, e.g. it distinguishes the linksin (1-2).1B. A digression on Artin–Tits groups.
Our motivation for this project, from which wehave now somewhat strayed, was to further our understanding of the connection betweenlow-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 ≥ ∪ {∞} . To any such diagram, we may associate the Artin–Titsgroup: AT(Γ) := (cid:10) β i , i ∈ V | . . . β i β j β i (cid:124) (cid:123)(cid:122) (cid:125) m ij factors = . . . β j β i β j (cid:124) (cid:123)(cid:122) (cid:125) m ij factors (cid:11) . (1-4)This group is an extension of the corresponding Coxeter group: W(Γ) := (cid:10) σ i , i ∈ V | σ i = 1 , . . . σ i σ j σ i (cid:124) (cid:123)(cid:122) (cid:125) m ij factors = . . . σ j σ i σ j (cid:124) (cid:123)(cid:122) (cid:125) m ij factors (cid:11) . (1-5)The jumping-off point is the classical observation that ℬ r( n ) = ℬ r(0 , n ) is isomorphic tothe Artin–Tits braid group of type A , while ℬ r(1 , n ) is isomorphic to the Artin–Tits groupof type C = B and extended affine type A . More-surprising is the lesser-known fact that DAVID E.V. ROSE AND DANIEL TUBBENHAUER ℬ r(2 , n ) is isomorphic to the Artin–Tits group of affine type C [All02]. The following tablesummarizes these known connections, details of which can be found in e.g. [All02, Section 4]and [Bri73].Genus type A type C g = 0 ℬ r( n ) ∼ = AT( A n − ) ? g = 1 ℬ r(1 , n ) ∼ = Z (cid:110)
AT(˜ A n − ) ∼ = AT(ˆ A n − ) ℬ r(1 , n ) ∼ = AT( C n ) g = 2 ? ℬ r(2 , n ) ∼ = AT(˜ C n ) g ≥ ? ? (1-6)Herein, A n − denotes the type A Coxeter diagram with n − nodes, while ˜ A n − denotes theaffine type A Coxeter diagram with n nodes and and ˆ A n − is the corresponding extendedaffine type. Similarly, C n and ˜ C n denote the type C = B and affine type C (but not affinetype 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’sconstruction of the HOMFLYPT polynomial, and the second row has similarly been exploitedin topological considerations, see e.g. [OR07] and [El18]. The type C column, however, hasreceived less attention, especially in the affine, g = 2 case, where not much appears to beknown about connections to link invariants. (However, this case has been explored froma representation-theoretic point of view, see e.g. [DR18].) A notable example is work ofGeck–Lambropoulou [GL97] in the g = 1 case, where a HOMFLYPT polynomial for links in ℋ 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 andaffine type C Hecke algebras and Soergel bimodules.By contrast, our construction in the present paper exploits the relation between ℬ r( g, n ) and ℬ r( g + n ) , and the fact that the latter is an Artin–Tits group. Indeed, our constructioncan be recast as follows. By viewing the distinguished strands as usual strands, we obtainan injective group homomorphism ℬ r( g, n ) (cid:44) → ℬ r( g + n ) . Since the latter is an Artin–Titsgroup, we can assign a complex of Soergel bimodules to any braid 𝒷 ∈ ℬ r( g, n ) . Now, beforetaking Hochschild cohomology (doing so immediately would give an invariant less-sensitiveto the topology of the handlebody), we glue on an additional Soergel bimodule that allowsinvariance under the Markov Theorem for ℬ r( g, n ) , but not for ℬ r( g + n ) . In fact, ourprocedure is slightly more general in that it uses an embedding of ℬ r( g, n ) into the coloredbraid group, and singular Soergel bimodules.1C. Future outlook.
In addition to our planned investigation in type C [RT], we believethere are a number of interesting future directions. • The relation between ℬ r( g, n ) and Hecke algebras. These exists a Hecke-likealgebra associated to ℬ r( g, n ) for general g , see e.g. [Lam00]. In the g = 0 , cases,this algebra matches the Hecke algebras associated to the Artin–Tits groups in thetype A column of (1-6). These algebras have not been widely studied, e.g. to ourknowledge 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 ℬ r( g, n ) given below in Definition 2.4 hintsto a connection to the Artin–Tits group associated to the Coxeter diagram that isobtained from the type A n diagram by adjoining g additional vertices. These verticesare attached to each other with ∞ -labeled edges, and to the first type A vertex with -labeled edges. (Something very similar was also observed in [Lam00, Remark 4].) OMFLYPT HOMOLOGY FOR LINKS IN HANDLEBODIES 5
For example, the g = 3 case is as follows:case g = 3 : · · · n − ∞ ∞∞ (1-7) Here, we depict k -labeled edges (for k < ∞ ) as k − unlabeled edges. In fact, ℬ r( g, n ) is a quotient of the associated Artin–Tits group, so one could hope to extractinvariants of 𝓁 ⊂ ℋ g from (a suitable quotient of) the corresponding Hecke algebraand/or Soergel bimodules. • Connections to algebraic geometry.
Work of Webster–Williamson [WW11] re-lates the Jones–Ocneanu trace on the type A Hecke algebra to the equivariant coho-mology of sheaves on SL n , and extends this to other types. It would be interesting toidentify geometry related to ℬ r( g, n ) and, more generally, links in ℋ g . One fertileavenue 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 arelation between the category of type A Soergel bimodules and the flag Hilbert schemeof C . The appearance of the latter can be interpreted as considering the closure of abraid 𝒷 ∈ ℬ r( n ) in the complement of an n -component unlink. Since the graph givingthe 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.Finally, another avenue of exploration is to extend various (known or conjectural)physical predications concerning ( g = 0 ) triply-graded homology to higher genus, see e.g. [GGS18] or [GS12], and [QRS18, Section 6.3] or [TVW17] for related results.1D. Conventions.
We now summarize various conventions used in this paper.
Convention . We work over an arbitrary field K of characteristic . This requirementis only needed in Section 4: the reader interested in integral versions of our results fromSection 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 andthe diagrammatic definitions differ when working integrally or in characteristic p .) All theresults from Section 3 then hold verbatim over Z . However, we do not currently have integralversions of the singular Soergel diagrammatics in Section 4. Convention . We will find it convenient to depict morphisms in certain categories (and -morphisms in certain -categories) diagrammatically. We will read such diagrams frombottom-to-top (and in the presence of a monoidal or -categorical structure, also right-to-left).These reading conventions are summarized by ABC ab cd (cid:33) (bd) ◦ (ac) : A → B → C . (1-8)Moreover, all such diagrams are invariant under (distant) height exchange isotopy (up toisomorphism, in the -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 notimportant for the argument in question. Convention . We will work with Z k -graded categories throughout, for k = 1 , , . The threegradings of importance are the internal degree q , the homological degree t (both appearingfrom Section 3 onward), and the Hochschild degree a (making its appearance in Section 4). DAVID E.V. ROSE AND DANIEL TUBBENHAUER
There are competing notions of what is meant by a graded category, so we now detailour conventions, focusing on the q -degree. Let C be a category enriched in Z -graded abeliangroups, i.e. for objects X and Y , Hom C ( X , Y ) is a Z -graded abelian group: Hom C ( X , Y ) = (cid:77) d ∈ Z Hom C ( X , Y ) d (1-9)Given such a category, we can introduce a formal grading-shift functor q and consider thecategory (cid:102) C q in which objects are given by formal shifts q s X of objects in C , and Hom (cid:102) C q ( q s X , q t Y ) = (cid:77) d ∈ Z Hom C ( X , Y ) d + t − s . (1-10) i.e. (cid:102) C q is again enriched in Z -graded abelian groups. Finally, we let C q be the category withthe same objects as (cid:102) C q , but where we restrict to q -degree zero morphisms, i.e. Hom C q ( q s X , q t Y ) = Hom C ( X , Y ) s − t . (1-11)Note that C q is not enriched in Z -graded abelian groups, but is equipped with an autoequiv-alence shift functor q . It is categories of this form that will be of primary interest in thiswork.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 HOM C q ( X , Y ) := (cid:77) d Hom C q ( q d X , Y ) (1-12)and we note that HOM C q ( q s X , q t Y ) = q t - s HOM C q ( X , Y ) , (1-13)where on the right-hand side the power q denotes a shift of the indicated Z -graded abeliangroup.Our consideration of Z - and Z -graded categories is analogous – in these cases we haveadditional shift functors t and a , and we restrict to t - and a -degree zero maps, unless otherwiseindicated. However, we will reserve the capitalization notation HOM when considering “graded
Hom s” with respect to the q -degree only.Lastly, we note that these considerations carry over to -categories as well, where the aboveapplies to the Hom -categories in our -category, i.e. to the - and -morphisms. Acknowledgments.
This project began during a visit of the first named author to theUniversität Zürich, and he thanks them for excellent working conditions. It continued duringvisits of both authors to the Kavli Institute for Theoretical Physics for the program QuantumKnot Invariants and Supersymmetric Gauge Theories. As such, this research was supportedin part by the National Science Foundation under Grant No. NSF PHY-1748958. We thankthem for the engaging and collaborative atmosphere. Some of the ideas for this project cameup during a discussion of the second author and Catharina Stroppel, and he thanks her forfreely sharing her ideas about HOMFLYPT homology outside of type A .Moreover, D.E.V.R. thanks Andrea Appel, Ben Elias, and Matt Hogancamp for usefuldiscussions, 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 andencouraging 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 financedthe first author’s visit to Zürich. OMFLYPT HOMOLOGY FOR LINKS IN HANDLEBODIES 7 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 -manifold with boundary obtained by attaching g -handles to the closed -ball ℋ . 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 ⊂ 𝒮 in the -sphere 𝒮 , i.e. the -manifoldgiven by the union of Σ g with the component of 𝒮 (cid:114) Σ g that does not contain the point atinfinity.We will typically work with another presentation for ℋ g , given by the closure in 𝒮 ofthe complement of an auxiliary handlebody ℋ cg . We view ℋ cg as consisting of g parallel -handles that are attached to the closure of a small neighborhood of ∞ ∈ 𝒮 . See the grayportion of the first figure in Example 2.2 for the g = 3 case. In order to connect with thecategorical representation theory used to produce our link invariant, we note that ℋ cg isisotopic 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 asbeing near the top of the core strands, hereby viewing ℋ cg as being obtained from the corestrands by first gluing on a graph with g ( g + 1) -valent vertices) and then taking their closure. See the gray portion of the second figurein Example 2.2.We will refer to this presentation of ℋ g = 𝒮 (cid:114) ℋ cg as the costandard presentation, andnote that it contains a copy of the standard presentation, to which it is isotopic, given byintersecting with a closed -ball that meets each core strand in a segment.We consider oriented links 𝓁 ⊂ ℋ g , which, in the costandard presentation, are given bylinks in 𝒮 (cid:114) ℋ cg . Equivalently, using the isotopy with the standard presentation, such linksare given by links in the closed -ball that avoid its intersection with the core strands. Finally,two links in 𝓁 , 𝓁 (cid:48) ⊂ ℋ g are isotopic, denoted by 𝓁 ∼ 𝓁 (cid:48) , if and only if the corresponding linksin 𝒮 (in the costandard presentation) are isotopic through an isotopy that keeps ℋ cg fixedpointwise.2B. Alexander’s theorem.
There is a corresponding notion of n strand braids in a genus g handlebody. Strictly speaking, we define ℬ r( g, n ) to be the braid group of the surface 𝒟 g givenas the complement of g disjoint open disks in the closed disk 𝒟 . The standard presentationof ℋ g may be identified with the product 𝒟 g × [0 , ⊂ 𝒮 , so braids in ℬ r( g, n ) “live in” ℋ g . The group ℬ r( g, n ) , which we call the handlebody braid group, can be equivalentlydescribed as follows. There is a subgroup of the classical braid group ℬ r( g + n ) on g + n strands consisting of braids that are pure on the first g strands, and a homomorphism fromthis subgroup to the classical braid group ℬ r( g ) given by forgetting the final n strands. Thekernel of this homomorphism is precisely ℬ r( g, n ) . (Informally, ℬ r( g, n ) consists of braids on g + n strands in which the first g strands do not braid among themselves.) Slightly abusingnotation, we will again refer to the first g strands of a braid in ℬ r( g, n ) as core strands; theycorrespond to the core strands above as we now describe.The handlebody braid group ℬ r( g, n ) is related to links in ℋ g in a manner parallelingthe relation between the classical braid group ℬ r( n ) and links in 𝒮 . That is, given a braid 𝒷 ∈ ℬ r( g, n ) , one obtains a link 𝒷 ⊂ ℋ g via a closure procedure as follows: the first g strandsin 𝒷 are joined at each of their ends to the point at infinity, and the remainder of the braid isclosed as in the classical case. In this way, we obtain a link 𝒷 ⊂ ℋ g where the closure of thelast n strands constitutes 𝒷 , and the first g strands in 𝒷 become the core strands in ℋ cg . Asin our discussion of ℋ cg 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 DAVID E.V. ROSE AND DANIEL TUBBENHAUER the top of the core strands. Specifically, the closure procedure consists of merging the g corestrands to meet the strand at infinity, then splitting the strand at infinity into g strands, andfinally taking the standard closure of all strands. Example . We have ℬ r(0 , n ) ∼ = ℬ r( n ) , which corresponds to links in the closed -ball ℋ ;we call this the classical case. In genus one, ℬ r(1 , n ) consists of all braids in ℬ r(1 + n ) thatare pure on the first strand, and ℋ is a solid torus. Example . Here we illustrate the closure procedure for 𝒷 ∈ ℬ r(3 , . • ∞ • ∞ 𝒷 𝒷ℋ c 𝒮 ∼ 𝒷 𝒷ℋ c 𝒮 𝒷 ∈ ℬ r(3 , , 𝒷 ⊂ ℋ (2-1)The braid itself is depicted as the solid strands in the indicated rectangle, while the dashededges correspond to the closure procedure described above. The thin, black components (bothsolid and dashed) give the link 𝒷 , while the thick, gray graph (again, both solid and dashed)depicts ℋ cg .The next result shows that, up to isotopy, all links in ℋ g arise from the closure procedurefor handlebody braids described above. The proof is analogous to the classical case. Theorem 2.3. (Alexander’s Theorem in a handlebody; [HOL02, Theorem 2] .) Given a link 𝓁 ⊂ ℋ g there exists a braid 𝒷 ∈ ℬ r( g, n ) such that 𝒷 ∼ 𝓁 ⊂ ℋ g . (cid:3) Generators and relations for braids in handlebodies.
We now recall the algebraicpresentation of ℬ r( g, n ) . Definition 2.4.
The group
Br( g, n ) is the group generated by 𝒷 , . . . , 𝒷 n − and 𝓉 , . . . , 𝓉 g ,called braid and twist generators, respectively, subject to the relations 𝒷 j 𝒷 i 𝒷 j = 𝒷 i 𝒷 j 𝒷 i if | i − j ] = 1 , 𝒷 j 𝒷 i = 𝒷 i 𝒷 j if | i − j ] > , (2-2) 𝒷 𝓉 i 𝒷 𝓉 i = 𝓉 i 𝒷 𝓉 i 𝒷 , 𝒷 i 𝓉 j = 𝓉 j 𝒷 i if i > , (2-3) (cid:0) 𝒷 𝓉 i 𝒷 − (cid:1) 𝓉 j = 𝓉 j (cid:0) 𝒷 𝓉 i 𝒷 − (cid:1) for i < j. (2-4)By convention, Br( g,
0) = { } , and we omit the twist generators when g = 0 and the braidgenerators when n = 1 .The following theorem identifies ℬ r( g, n ) and Br( g, n ) , and we likewise do for the duration.In particular, we identify Br( n ) = Br(0 , n ) and ℬ r( n ) . Proposition 2.5. ( [Ver98, Theorem 1] & [Lam00, Section 5] .) There is an isomorphism ofgroups (cid:3) (2-5) Br( g, n ) ∼ = ℬ r( g, n ) . OMFLYPT HOMOLOGY FOR LINKS IN HANDLEBODIES 9
An explicit isomorphism realizing Proposition 2.5 is given on the braid and the twistgenerators as follows: 𝒷 i (cid:55)→ i+1ii i+1 ... ... ... & 𝓉 i (cid:55)→ ...... ...... ... (2-6)The inverse of 𝒷 i is given, as usual, by the corresponding opposite crossing, and the inverseof 𝓉 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). Example . Under this map, the first relation in (2-2) corresponds to the braid-like Reide-meister III relation, and the second relations in (2-2) and (2-3) correspond to planar isotopy.The four term relation in (2-3) and the relation in (2-4) become e.g. the relations (2-7) and(2-8), respectively: 𝒷 𝓉 i 𝒷 𝓉 i = 𝓉 i 𝒷 𝓉 i 𝒷 (cid:33) ...... = i 1i 1 ...... (2-7) (cid:0) 𝒷 𝓉 i 𝒷 − (cid:1) 𝓉 j = 𝓉 j (cid:0) 𝒷 𝓉 i 𝒷 − (cid:1) (cid:33) i 1ji j 1 ...... ...... = i j 1i jj 1 ...... ...... (2-8)The non-trivial statement in Proposition 2.5 is that these relations are sufficient.2D. Markov’s theorem.
Let ℬ r( g, ∞ ) := (cid:116) n ∈ N ℬ r( g, n ) , the set of all braids in ℋ g . Definition 2.7.
Let ℬ r( g, ∞ ) Ma := ℬ r( g, ∞ ) / ∼ be the quotient given by conjugation (2-9)in (each) ℬ r( g, n ) by elements 𝓈 ∈ (cid:104) 𝒷 , . . . , 𝒷 n − (cid:105) and stabilization (2-10), i.e. 𝒷 ∼ 𝓈𝒷𝓈 − for 𝒷 ∈ ℬ r( g, n ) , 𝓈 ∈ (cid:104) 𝒷 , . . . , 𝒷 n − (cid:105) (cid:33) ... ... n ... ... n 𝒷 ∼ ... ... n ... ... n 𝒷 𝓈𝓈 -1 (2-9) ( 𝒸 ↑ ) 𝒷 n ( 𝒷 ↑ ) ∼ 𝒸𝒷 ∼ ( 𝒸 ↑ ) 𝒷 − n ( 𝒷 ↑ ) for 𝒷 , 𝒸 ∈ ℬ r( g, n ) , (cid:33) nn 𝒷𝒸 ...... ...... ∼ nn 𝒷𝒸 ...... ...... ∼ nn 𝒷𝒸 ...... ...... (2-10)where 𝒷 ↑ ∈ ℬ r( g, n + 1) is the braid obtained from 𝒷 ∈ ℬ r( g, n ) by adding a strand to theright. Remark . The conjugation (2-9) is weaker than in the classical case – there one canconjugate by any element, instead of just by certain elements. This will play an importantrole in our construction of the HOMFLYPT invariant, see Proposition 4.8. On the otherhand, 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.
Theorem 2.9. (Markov’s Theorem in a handlebody; [HOL02, Theorem 5] .) Let 𝒷 , 𝒸 ∈ ℬ r( g, ∞ ) , then 𝒷 ∼ c ⊂ ℋ g if and only if 𝒷 = 𝒸 ∈ ℬ r( g, ∞ ) Ma . (cid:3) Remark . Although it may appear that Definition 2.7 omits conjugation by certainelements that clearly give isotopic closures e.g. by the “maximal loop” ω = 𝓉 g . . . 𝓉 or itsinverse, [HOL02, Section 5] shows how conjugation by such elements can be described interms of the above Markov moves.2E. From handlebody braids to classical braids.
Recall from Section 1B that one ofour main ingredients in constructing homological invariants of links in ℋ g for all g ≥ is therelation between ℬ r( g, n ) and (a colored variant of) the type A braid group ℬ r( g + n ) : Wehave a group homomorphism ℬ r( g, n ) → ℬ r(0 , g + n ) given by viewing the core strands as“usual” strands, e.g. gg 1ii 1 ...... (cid:55)→ gg g+1ii g+1 ...... (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 (cid:3) (2-12) ℬ r( g, n ) (cid:44) → ℬ r( g + n ) . However, Proposition 2.11 is only one ingredient in our construction, since invariance underthe procedure ... ... n ... ... n 𝒷 (cid:55)→ ... ... g+n ... ... g+n 𝒷 ∼ ... ... g+n ... ... g+n 𝒷𝓈𝓈 -1 ← (cid:91) ... ... n ... ... n 𝒷𝓈𝓈 -1 (2-13)(i.e. under “conjugation in ℬ r( g + n ) ”) is not desirable for an invariant of 𝓁 ∼ 𝒷 ⊂ ℋ g , cf. Remark 2.8. As such, we will use the theory of singular Soergel bimodules to mimic themerging and splitting of the core strands in the closure procedure for ℬ r( g, n ) , which willlead to invariants of 𝒷 that are not invariant under (2-13).3. Braids in handlebodies and singular type A Soergel bimodules
In the present section, we construct a map from ℬ r( g, n ) to the -category of singularSoergel bimodules.3A. Parabolic subgroups and Frobenius extensions.
Fix N ∈ N ≥ and let R := R N := K [ x , . . . , x N ] be the q -graded polynomial ring with q deg( x i ) = 2 for all i (by convention, R := K ). The symmetric group S( N ) = W( A N − ) acts on R via σ i · x j = x i +1 if i = j, x i if i = j + 1 , x j else . (3-1) Remark . Recall that Tits defined a faithful representation of any Coxeter group
W(Γ) on a real vector space of dimension | V | , commonly called the reflection representation of W(Γ) . (Recall our notation from Section 1B.) This representation is a crucial ingredient inthe original definition of the associated category of Soergel bimodules, see [Soe92, Section1.4]. In our case, this is the standard (irreducible) representation of S( N ) of dimension N − .By contrast, the representation given by (3-1) is built from the N -dimensional permutationrepresentation, which decomposes as a direct sum of the standard representation and the OMFLYPT HOMOLOGY FOR LINKS IN HANDLEBODIES 11 trivial representation. By e.g. [EK10, Section 4.6] and [EL17, Theorem 2.7 and Proposition2.10], the difference (akin to the difference between considering gl N rather than sl N ) will notplay a role in the present work, in the sense that all results from the cited literature hold inthis case as well.Fix any tuple I = ( k , . . . , k r ) ∈ N r ≥ with k + · · · + k r = N . (Note further that choosing I also determines N since N = k + · · · + k r . We will tacitly use this throughout.) By definition,the corresponding parabolic subgroup is S I ( N ) := S( k ) × · · · × S( k r ) ⊂ S( N ) . (3-2)Since there is a bijection between tuples and parabolic subgroups, we will implicitly identifythem, e.g. I ⊂ J denotes an inclusion of parabolic subgroups.Given a parabolic subgroup I , we let R I := R S I ( N ) be the ring of invariants. This ring is q -graded, since the action in (3-1) is q -homogeneous. Example . The parabolic subgroups in (3-2) can alternatively be defined by choosingcorresponding subsets of the vertices
V = { , . . . , N − } of the type A N − Coxeter diagram(with the left-right order of the vertices). For type A one gets (1 , , , (cid:33) ∅ , (2 , , (cid:33) { } , (1 , , (cid:33) { } , (1 , , (cid:33) { } , (3 , (cid:33) { , } , (2 , (cid:33) { , } , (1 , (cid:33) { , } , (4) (cid:33) { , , } . (3-3)Above we have listed all choices of tuples and the associated parabolic subgroups. Thus, R (1 , , , = R ∅ is R itself, while R (4) = R { , , } is the K -algebra of symmetric polynomials infour variables.For the duration, we will use the following ordering convention for parabolic subgroups I , J , K , L and their rings of invariants: JI LK ⊂ ⊂ ⊂ ⊂ (cid:33) R J R I R L R K ⊃ ⊃ ⊃ ⊃ .(3-4)The q -degree inclusion ι JI : R J (cid:44) → 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 . Inthe present case, the latter is built using the Demazure operators ∂ σ i : R → R { i } ⊂ R , givenby ∂ σ i ( f ) := ( f − σ i · f ) /α i for the roots α i = x i − x i +1 . The collection { ∂ σ i } satisfies theclassical 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 ∂ JI : R I → R J , f (cid:55)→ ∂ w I w − J ( f ) (3-6)and is of q -degree (cid:96) ( I ) − (cid:96) ( J ) . Here w I is the longest element in S I ( N ) , and (cid:96) ( I ) denotesits 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 thedefinition of the latter below). To wit, given a basis { a i } for R I over R J , we can find adual basis { a (cid:63)i } satisfying ∂ JI ( a i a (cid:63)j ) = δ ij . Given this, we obtain the Frobenius element a := (cid:80) i a i ⊗ a (cid:63)i , which is of q -degree (cid:96) ( J ) − (cid:96) ( I ) and independent of the choice of { a i } .This gives multiplication and comultiplication maps µ JI : R I ⊗ R J R I → R I , f ⊗ g (cid:55)→ f g, q -degree , ∆ JI : R I → R I ⊗ R J R I , f (cid:55)→ f a , q -degree (cid:96) ( J ) − (cid:96) ( I )) . (3-7)These morphisms of bimodules are unital and counital with respect to ι JI and ∂ JI , respectively. Example . For I = ∅ and J = { i } , we have w I = 1 and w J = σ i . It follows that { , α i } and { (cid:63) = α i , ( α i ) (cid:63) = 1 } are dual bases for R as an R { i } -module, and a = (1 ⊗ α i + α i ⊗ .Finally, for later use, let us explicitly identify the rings R I for all I = ( k , . . . , k r ) . To thisend, 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 (cid:2) e ( X ) , . . . , e k ( X ) (cid:3) ⊗ · · · ⊗ K (cid:2) e ( X r ) , . . . , e k r ( X r ) (cid:3) , (3-8)where e j ( X i ) denotes the j th elementary symmetric function in the variables X i . Note that q deg( e j ) = 2 j . In particular, q dim K (R I ) = r (cid:89) j =1 k j (cid:89) i =1 11 − q i . (3-9)3B. A reminder on type A singular Soergel bimodules. We now briefly recall thecategory of singular Soergel bimodules SS q ( N ) = SS q ( A N − ) of type A N − , which categorifiesthe Hecke/Schur algebroid of type A [Wil11, Theorem 1.2] in characteristic . Details (inmore generality) can be found e.g. in [Wil11], or [EL17] and [ESW17] for the underlyingdiagrammatic calculus.Define the merge (“restriction”) and split (“induction”) bimodules as follows: J M I := q (cid:96) ( I ) − (cid:96) ( J ) R J ⊗ R J R I , I S J := R I ⊗ R J R J , (3-10)where we follow the conventions from (3-4). Here, we have indicated the left/right actionsusing 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, whichwe 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 bimodule isomorphisms that we implicitly use below: L M J M I ∼ = L M I ∼ = L M K M I , I S J S L ∼ = I S L ∼ = I S K S L . (3-11)All of the isomorphism in (3-11) are essentially identities, as the careful reader is invited tocheck. (Note e.g. that f ⊗ g ⊗ h = 1 ⊗ ⊗ f gh ∈ L M J M I .) Definition 3.4.
Let SS q ( N ) be the K -linear, q -graded -category given as the additiveKaroubi -closure (meaning taking direct sums and summands) of the -category whereobjects are parabolic subgroups I ⊂ S( N ) , -morphisms are generated by q -shifts of R I : I → I , J M I : I → J , and I S J : J → I (3-12)for I ⊂ J , and -morphisms are (all) bimodule maps of q -degree . Example . We have q - R ⊗ R { i } R ∼ = , S M , , 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 − can be identified withthe -category End SS q ( N ) ( ∅ ) , which has just one object (hence is a monoidal category). OMFLYPT HOMOLOGY FOR LINKS IN HANDLEBODIES 13
Web diagrammatics.
Following e.g. ideas in [MSV11, Section 3], the generating -morphisms from (3-12), and compositions thereof, admit a description in terms of anMOY-type calculus, which we now sketch.The basic building blocks are the identity, merge, and split bimodules, which are depictedusing the following (local) graphical notation: k i k i (cid:33) R k i , k lk + l (cid:33) k + l M k,l , k lk + l (cid:33) k,l S k + l . (3-13)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 tocomposition of -morphisms, and e.g. composing on the left corresponds to stacking onthe top. Moreover, we can place such diagrams side-by-side, which corresponds to takingthe tensor product over K . Hence, we can associate a singular Soergel bimodule to eachtrivalent graph that we can build from these diagrams via these operations. (Note thatwebs corresponding to singular Soergel bimodules never have edges of negative label, butwe will allow them in formulas for convenience of notation, with the understanding that thecorresponding bimodules are zero.) Example . 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: := ⊗ R (cid:33) q - R ⊗ R { i } R ∼ = , S M , . (3-14)Also of importance will be the ladder-rung bimodules: k la := lk a & k la := k la (3-15)that will be used to build the square bimodules appearing in the complexes in (3-18) below. Example . There exist q -degree bimodule isomorphisms k + l + mk l m ∼ = k + l + mmlk & k + l + mk l m ∼ = k + l + mmlk (3-16)that follow from the isomorphisms in (3-11). Hence, we can unambiguously write k k r k + . . . + k r ... & k k r k + . . . + k r ... (3-17)3D. Rickard–Rouquier complexes.Definition 3.8.
Given an additive category C , we denote its bounded homotopy categoryby K b (cid:0) C (cid:1) . This is the category whose objects are bounded chain complexes, and whosemorphisms are homotopy classes of chain maps. We will use (cid:39) to denote isomorphisms in K b (cid:0) C (cid:1) , i.e. homotopy equivalence.Recalling Section 1D, we can view the objects in K b (cid:0) C (cid:1) as finite direct sums (cid:76) i t k i X i ,equipped with a differential d with t deg( d ) = − . There is a t -degree zero inclusion ofcategories C (cid:44) → K b (cid:0) C (cid:1) given by considering objects of C as one-term complexes concentratedin t -degree . We also remark that we can consider K b (cid:0) (cid:67) (cid:1) for a -category (cid:67) , by passing to the homotopy category in each Hom -category. In particular, if C is monoidal, then so is K b (cid:0) C (cid:1) .We now recall Rickard–Rouquier complexes, i.e. complexes of singular Soergel bimodulesthat determine maps from the (colored) braid group(oid) into certain Hom -categories in K b (cid:0) SS q ( N ) (cid:1) . Our terminology here arises as these complexes correspond to the Rickardcomplexes (originally defined for symmetric groups) in categorified quantum groups, but alsoagree with the type A Rouquier complexes in the “uncolored” k = l = 1 case.They are given as follows: llk k := kl lk d +0 −−→ tq - kl l k d +1 −−→ . . . d + m − −−−→ t m q - m kl lmkk kll := t - m q m kl lmk d − m − −−−→ . . . d − −−→ t - q kl l k d − −−→ kl lk (3-18)where m = min( k, l ) . Our notation denotes e.g. that, as a tq -graded bimodule, (cid:74) β i (cid:75) k,l is thedirect sum of the indicated terms, and the arrows depict the non-zero components of thedifferentials. Recalling the bimodule maps from (3-11), (3-5), (3-6), (3-7), and omitting the tq -shifts, these are given by d + i : kl lki kl lk i kl lk i kl lk i kl lki + ∂ι (3-11)(3-11) µ ∆ ι∂ : d − i (3-19)Here the corresponding parabolic subsets, which determine the bimodule maps, can be readfrom the indicated sequence of webs, and we use e.g. ι JI as ι JI : R J ∼ = R J ⊗ R J R J ⊗ R J R J (cid:44) → R J ⊗ R J R I ⊗ R J R J = J M I S J . (3-20) Remark . We note that the differential in the Rickard–Rouquier complexes can be de-scribed diagrammatically using type A singular Soergel calculus, see e.g. [EL17, Section2]. Alternatively, we could work with the n → ∞ limit of the -category of gl n foams todescribe these -morphisms in SS q ( N ) (here, n is a parameter independent of N ). In fact,these two descriptions are equivalent, as the type A singular Soergel calculus corresponds tothe “calculus of seams” in the foam framework. (See e.g. [QRS18, Section 5.2] for a precisestatement.)Finally, the fact that these indeed are complexes follows e.g. by comparing (3-18) to theRickard complex in the categorified quantum group, as in Remark 3.9. Example . In the uncolored case k = l = 1 the complexes are
111 1 =
111 1 µ , −−→ tq - & = t - q ∆ , −−−→
111 1 (3-21)
Remark . The conventions in Example 3.10 are the same as in [Rou06], except that inthat work, there is no shift on the bimodule , S M , . OMFLYPT HOMOLOGY FOR LINKS IN HANDLEBODIES 15
Example . There exist q -degree isomorphisms in K b (cid:0) SS q ( N ) (cid:1) l ml + mk k (cid:39) l ml + mk k & k + llk (cid:39) q kl k lk + l & k + lk l (cid:39) q - kl k lk + l (3-22)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 correspondingtuple ( i.e. for an r -tuple I , I = r ). Given a braid generator 𝒷 i ∈ ℬ r( I ) , we let (cid:113) 𝒷 ± i (cid:121) I denote the complex given by placing appropriately labeled vertical strands next to thecorresponding complex in (3-18), i.e. by taking tensor product over K with the rings R ( k ,...,k i − ) and R ( k i +2 ,...,k I ) . Definition 3.13.
For I and 𝒷 ∈ ℬ r( I ) , fix an expression 𝒷 = 𝒷 ± i . . . 𝒷 ± i r . Define (cid:74) 𝒷 (cid:75) I := (cid:113) 𝒷 ± i (cid:121) I (cid:48) . . . (cid:113) 𝒷 ± i r (cid:121) I where, on the right-hand side, we use composition in K b (cid:0) SS q ( N ) (cid:1) , i.e. tensor product of thecomplexes of singular Soergel bimodules.By e.g. the results in [QRS18, Section 5.2], the complex (cid:74) 𝒷 (cid:75) I does not depend, up toisomorphism, on the choice of expression for 𝒷 . Thus, the assignment 𝒷 (cid:55)→ (cid:74) 𝒷 (cid:75) I gives anaction of ℬ r( I ) on K b (cid:0) SS q ( N ) (cid:1) . We get: Proposition 3.14.
There is an action of ℬ r( g, n ) on K b (cid:0) SS q ( N ) (cid:1) determined by the as-signment 𝒷 (cid:55)→ (cid:74) 𝒷 (cid:75) I .Proof. By the discussion above, we have an action of the classical braid group. Composingthis action with the map from Proposition 2.11 gives the desired action of the handlebodybraid group. (cid:3) Colored HOMFLYPT homology for links in handlebodies
In this section, we proceed to construct our triply-graded invariant of links in ℋ g , withTheorem 4.7 and Corollary 4.13 being the main statements. We keep the notation from theprevious sections and begin with some preliminaries.4A. A reminder on Hochschild cohomology.
Let A be a q -graded K -algebra, and recallthat we may regard any q -graded A -bimodule B as a q -graded left module over the envelopingalgebra A ⊗ A op . The Hochschild cohomology of A with coefficients in B is the aq -graded K -vector space HH • (A , B) := (cid:77) a ∈ Z HH a (A , B) (4-1)with a -degree component defined by HH a (A , B) := EXT a A ⊗ A op (A , B) = (cid:77) s ∈ Z Ext a A ⊗ A op ( q s A , B) . (4-2)(Compare our notation here to Convention 1.3.)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 constructedusing 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 homologyand cohomology are isomorphic (up to a shift). We work with the latter since e.g. in thisframework the invariant of the (colored) unknot inherits a natural algebra structure [Hog18],which is important for various considerations. Example . Let I = ( k , . . . , k r ) . Recall that R I is a polynomial ring (and, in particular, isKoszul). 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 r (cid:79) j =1 k j (cid:79) i =1 (cid:0) hq i R I ⊗ R I e i ⊗ − ⊗ e i −−−−−−−→ R I ⊗ R I (cid:1) . (4-3)Here h denotes a shift up in an auxiliary homological degree, and the outer tensor productsare taken over R I ⊗ R I . Given a R I -bimodule B , taking the “internal” q -graded Hom ofcomplexes
HOM • R I ⊗ R I ( − , B) ( i.e. applying HOM R I ⊗ R I ( − , B) to the terms and differentialsof a chain complex to obtain a cochain complex) gives a complex concentrated in non-negativecohomological degree a , which is the negative of the h -degree. The a th cohomology of thiscomplex is HH a (R I , B) .Computing for B = R I gives the following. For each j , fix a set of variables { θ i | ≤ i ≤ k j } with aq deg( θ i ) = (1 , − i ) , and recall that aq deg( e i ) = (0 , i ) . We then have an isomorphismof aq -graded K -vector spaces HH • (R I , R I ) ∼ = r (cid:79) j =1 (cid:0) K [ e , . . . , e k j ] ⊗ (cid:86) • { θ i | ≤ i ≤ k j } (cid:1) , (4-4)where (cid:86) • { θ i | ≤ i ≤ k j } denotes the exterior algebra.Since Hochschild cohomology is functorial with respect to bimodule morphisms, we canapply HH • to a complex of R I -bimodules term-wise to obtain a complex of aq -graded K -vector spaces. (In fact, since our ring is commutative, these K -vectors spaces inherit an actionof 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 (cid:0) R I Bim q (cid:1) be its homotopy category. We get a functor HH • I ( − ) := (cid:77) a ∈ Z HH a I ( − ) : K b (cid:0) R I Bim q (cid:1) → K b (cid:0) K Vec aq (cid:1) (4-5)whose a -degree component is the functor HH a I ( − ) := HH a (R I , − ) : K b (cid:0) R I Bim q (cid:1) → K b (cid:0) K Vec q (cid:1) . (4-6)4B. Towards handlebody HOMFLYPT homology.
Fix integers
M, l , . . . , l n ∈ N ≥ ,called the core and link colors, respectively. For any g ≥ , these choices determine a parabolicsubset M := ( M, . . . , M, l , . . . , l n ) with M = g + n . We view M as providing a coloring forbraids 𝒷 ∈ ℬ r( g, n ) as in Section 3D, where strands are colored at the bottom by the entriesof M . We will call a colored braid ( 𝒷 , 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 anybraid or link will be colored by default. Example . The prototypical example of a balanced coloring is the case where the link isuncolored, i.e. where l = · · · = l n = 1 and M is arbitrary. In general, M should be viewedas being “very large,” i.e. M (cid:29) l i for all i ; compare e.g. to [ILZ18], where the core of thesolid torus is colored by a Verma module. Remark . It is possible to work with any balanced coloring of 𝒷 ∈ ℬ r( g, n ) . However, thecore 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 toour previous notation for this operation, which was simply concatenation.Recalling Example 3.7 and Proposition 3.14, and motivated by Remark 2.8, we define: OMFLYPT HOMOLOGY FOR LINKS IN HANDLEBODIES 17
Definition 4.4.
For 𝒷 ∈ ℬ r( g, n ) and ( 𝒷 , M ) a balanced coloring, we let (cid:74) 𝒷 (cid:75) ℋ g := M MM M l l l n l n ...... ... ⊗ R M (cid:74) 𝒷 (cid:75) M ∈ K b (cid:0) SS q ( M ) (cid:1) . (4-7) Example . In the cases g = 0 , we have (cid:74) 𝒷 (cid:75) ℋ g = (cid:74) 𝒷 (cid:75) M . For g = 2 , we have 𝒷 = & (cid:74) 𝒷 (cid:75) M = MM lMM l & (cid:74) 𝒷 (cid:75) ℋ = M lMM M l (4-8)For g ≥ , we generally have (cid:74) 𝒷 (cid:75) ℋ g (cid:54)(cid:39) (cid:74) 𝒷 (cid:75) M , cf. Proposition 4.8 below.
Definition 4.6.
For 𝒷 ∈ ℬ r( g, n ) and ( 𝒷 , M ) a balanced coloring we let HH • ℋ g ( 𝒷 ) := (cid:77) a ∈ Z HH a (cid:0) (cid:74) 𝒷 (cid:75) ℋ g (cid:1) . (4-9)By Proposition 3.14, HH • ℋ g ( 𝒷 ) is an invariant of the colored braid 𝒷 ∈ ℬ r( g, n ) takingvalues in K b (cid:0) K Vec aq (cid:1) .4C. Colored handlebody HOMFLYPT homology.
In (4-46) below we use HH • ℋ g ( 𝒷 ) to define HHH • ℋ g ( 𝒷 ) , an invariant of the colored link 𝒷 ⊂ ℋ g , valued in K Vec atq . To do so,we establish the following.
Theorem 4.7.
The assignment ℬ r( g, n ) → K b (cid:0) K Vec aq (cid:1) , 𝒷 (cid:55)→ HH • ℋ g ( 𝒷 ) (4-10) is invariant under the conjugation (2-9) and stabilization (2-10) relations for ℬ r( g, n ) , upto homotopy equivalence and grading normalization. Moreover, it is not generally invariantunder the classical conjugation relation (2-13) for ℬ r( g + n ) . The remainder of this section constitutes a proof of this theorem. Namely, invarianceunder conjugation holds as a special case of the corresponding result for colored, triply-gradedlink homology in 𝒮 , and Lemma 4.12 establishes invariance (up to a grading shift) understabilization. Proposition 4.8 shows the failure of invariance under the classical conjugationrelation. We stress the importance of this latter fact: any invariant of the classical braidgroup ℬ r( g + n ) , that additionally is invariant under classical conjugation and stabilization,gives rise to invariants of links in ℋ g using the inclusion ℬ r( g, n ) (cid:44) → ℬ r( g + n ) . However,such invariants are less-sensitive to the topology of ℋ g , as our results show. Proposition 4.8.
For g > and n > , there exists handlebody braids 𝒷 , 𝒷 (cid:48) ∈ ℬ r( g, n ) thatare conjugate in ℬ r( g + n ) , but satisfy HH • ℋ g ( 𝒷 ) (cid:54)(cid:39) HH • ℋ g ( 𝒷 (cid:48) ) . This shows that our handlebody homology, which is defined below to be the cohomologyof a renormalization of HH • ℋ g ( − ) , distinguishes these handlebody links, while the invariantobtained by including into ℬ r( g + n ) and using the classical (colored) triply-graded homologydoes not. Proof.
It suffices to give an example, and we provide one in the g = 2 and n = 1 case thatimmediately generalizes to any g ≥ and n ≥ . Let 𝒷 = 𝓉 𝓉 and 𝒷 (cid:48) = 𝓉 𝓉 , which areconjugate braids in ℬ r( g + n ) . We claim that HH • ℋ g ( 𝒷 ) (cid:54)(cid:39) HH • ℋ g ( 𝒷 (cid:48) ) , and exhibit thisexplicitly in the case that M = 1 . Indeed, if they were homotopy equivalent, then the Euler characteristics ( i.e. alternatingsums of aq -graded dimensions) of these complexes would agree. However, since the categoryof (usual) type A Soergel bimodules categorifies the type A Hecke algebra, and Hochschildcohomology categorifies the Jones–Ocneanu trace, this would imply that the Jones–Ocneanutraces of the following braided, trivalent graphs agree: 𝒷 & 𝒷 (cid:48) (4-11)Using the decategorification of the first equation in Example 3.10, this in turn would implythat the HOMFLYPT polynomials of the links given as the closures of 𝒷 & 𝒷 (cid:48) (4-12)agree. However, a computation shows that the difference between their (reduced) HOMFLYPTpolynomials is ( a − a - ) − ( q − q - ) , where a , q are variables (at the decategorified level)corresponding to a , q . (cid:3) We now turn out attention to the behavior of HH • ℋ g ( 𝒷 ) under stabilization (2-10). Ourmain technical tool will be the partial Hochschild trace from [Hog18, Section 3], which we nowadapt to the colored setting. The construction of this functor is motivated as follows. SinceHochschild cohomology satisfies the classical conjugation relation ( i.e. the relation (2-13)), weinformally view this operation as a mean to take the closure of (the singular Soergel bimoduleassociated to) a web diagram. In order to study the stabilization relation, we would like tobe able to take this closure “one strand at a time” in a manner that is compatible with takingHochschild cohomology.Recall that the Hochschild cohomology of an R I -bimodule M is defined as HH a (R I , M) = EXT a R I ⊗ R I (R I , M) ∼ = HOM D b (R I Bim q ) (R I , h a M) , (4-13)where here we follow Convention 1.3 for the q -graded Hom . Here D b (cid:0) R I Bim q (cid:1) is the boundedderived category, and we emphasize that the homological degree therein is not the t -degreefrom 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 discussionabove suggests that we should consider functors between the categories D b (cid:0) R I Bim q (cid:1) for various I that are compatible with the functors HOM D b (R I Bim q ) (R I , − ) .To this end, given I = ( k , . . . , k r ) we let I − := ( k , . . . , k r − ) , i.e. I − is obtained from I by removing the last entry. Let Q k r I := R I ⊗ R I (cid:14) ( e i ( X r ) ⊗ − ⊗ e i (cid:0) X r ) (cid:1) k r i =1 . Using thenotation in (3-8), we have R I ∼ = Q k r I ⊗ R I − ⊗ R I − R I − , (4-14)which suggests that we consider the functor I I : D b (cid:0) R I − Bim q (cid:1) → D b (cid:0) R I Bim q (cid:1) given byderived tensor product ⊗ L with Q k r I over R I − ⊗ R I − . We then obtain T I : D b (cid:0) R I Bim q (cid:1) →D b (cid:0) R I − Bim q (cid:1) , which we define to be the right adjoint to I I , using derived tensor-homadjunction. OMFLYPT HOMOLOGY FOR LINKS IN HANDLEBODIES 19
The functors T I and I I admit the following explicit descriptions. We have an isomorphism Q k r I ∼ = k r (cid:79) i =1 (cid:16) hq i R I ⊗ R I e i ⊗ − ⊗ e i −−−−−−−→ R I ⊗ R I (cid:17) =: K k r I (4-15)in D b (cid:0) R I Bim q (cid:1) , where the (outer) tensor product is taken over R I ⊗ R I . Since K k r I is acomplex of free R I ⊗ R I -modules, given any complex M ∈ D b (cid:0) R I Bim q (cid:1) , T I (M) is the complex HOM • R I ⊗ R I (K k r I , M) ∼ = k r (cid:79) i =1 (cid:16) M e i ⊗ − ⊗ e i −−−−−−−→ aq - i M (cid:17) . (4-16)(In the case that M is a complex, we interpret the latter as a shift of the cone of the indicatedchain map.) Similarly, for N ∈ D b (cid:0) R I − Bim q (cid:1) , we have that I I (N) = K k r I ⊗ R I − ⊗ R I − N , (4-17)where we again interpret the latter as the total complex of this double complex. Here, since K k r I ∼ = Q k r I and the latter is a free R I − -bimodule, we also have I I (N) = Q k r I ⊗ R I − ⊗ R I − N . (4-18)Our next result collects the salient features of I I and T I needed for our considerations,both of which follow immediately from the definition of I I and T I . Lemma 4.9.
For all M ∈ D b (cid:0) R I Bim q (cid:1) and all a ∈ Z , there is a functorial isomorphism HH a (R I , M) ∼ = HH a (cid:0) R I − , T I (M) (cid:1) . (4-19) Additionally, given N , P ∈ D b (cid:0) R I − Bim q (cid:1) , we have T I (cid:0) I I ( N ) ⊗ L R I M ⊗ L R I I I (P) (cid:1) ∼ = N ⊗ L R I T I (M) ⊗ L R I P (4-20) Finally, setting I := { k , . . . , k s } and I := { k s +1 , . . . , k r } , we have T I (cid:0) M ⊗ K M (cid:1) ∼ = M ⊗ K T I (M ) . (4-21) for M ∈ D b (cid:0) R I Bim q (cid:1) and M ∈ D b (cid:0) R I Bim q (cid:1) . 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 , fully faithful inclusion functor SS q ( I ) (cid:44) → D b (cid:0) R I Bim q (cid:1) givenby 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) sincesingular Soergel bimodules are free as either left or right R I -modules. (The latter can bededuced 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 thesingular setting. Since our eventual aim is to apply these results to HH • ℋ g ( − ) , we will focuson the (bimodules appearing in the) complex (cid:74) 𝒷 (cid:75) ℋ g . Let M := ( M, . . . , M, l , . . . , l n ) , then,for B ∈ SS q ( M − ) and C ∈ SS q ( M ) , we depict I M and T M as follows: I M B ...... ...... = B ...... ...... l n l n & T M C ...... ...... = C ...... ...... l n (4-22) 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 sameresult whether we close all strands at once or one at a time, while the second and third are l n NMP ...... ...... ∼ = l n NMP ...... ...... M M ... ......... ...... l n ∼ = M ... ......... ⊗ K M ...... l n (4-23)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 coloredpartial trace, and since we anticipate applications of this formula to explicit computations ofour invariant, we take the opportunity to extend loc. cit. to the colored setting.) Lemma 4.10.
For k, l ≥ , there is an atq -degree isomorphism kk l ∼ = l (cid:89) i =1 q k + aq - k - i − q i kk (4-24) Proof.
In the k = 0 case, the result simply claims that the l -colored circle is a K -vector spaceof aq -graded dimension (cid:81) li =1 1+ aq - i − q i . This follows directly from Example 4.1.We thus assume that k ≥ , 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 derivedcategory.To this end, we assign alphabets of q -degree variables to the boundary points of thecorresponding web as follows: X (cid:48) X (cid:48) X X (4-25)where X = k = X (cid:48) and X = l = X (cid:48) . Precisely, by this assignment we identify the(singular) Bott–Samelson bimodule k,l S k + l M k,l with the following quotient of the (shifted)polynomial ring generated by the elementary symmetric functions in these alphabets: q - kl K (cid:2) e r ( X ) , e r ( X (cid:48) ) , e s ( X ) , e s ( X (cid:48) ) (cid:3)(cid:46)(cid:0) e t ( X ∪ X ) − e t ( X (cid:48) ∪ X (cid:48) ) (cid:1) . (4-26)Here r, s, t are indices ranging ≤ r ≤ k , ≤ s ≤ l and ≤ t ≤ k + l ( i.e. we slightly abusenotation and let e r ( X ) denote e ( X ) , . . . , e k ( X ) , etc. ). The latter is quasi-isomorphic tothe object in D b (cid:0) R k,l Bim q (cid:1) given by the dg algebra K := q - kl K (cid:2) e r ( X ) , e r ( X (cid:48) ) , e s ( X ) , e s ( X (cid:48) ) (cid:3) ⊗ K (cid:86) • { θ t } , (4-27)where aq deg( θ t ) = ( − , t ) and d ( θ t ) = e t ( X ∪ X ) − e t ( X (cid:48) ∪ X (cid:48) ) . Computing partial tracethen gives that T k,l (K) ∼ = a l q - kl q - l ( l + K (cid:2) e r ( X ) , e r ( X (cid:48) ) , e s ( X ) , e s ( X (cid:48) ) (cid:3) ⊗ K (cid:86) • { θ t , ξ s } (4-28) OMFLYPT HOMOLOGY FOR LINKS IN HANDLEBODIES 21 where aq deg( ξ s ) = ( − , s ) and d ( ξ s ) = e s ( X ) − e s ( X (cid:48) ) .Since the right-hand side of (4-24) is quasi-isomorphic to a direct sum of copies of theKoszul complex associated to the elements e r ( X ) − e t ( X ) (cid:48) , we now aim to change variablesin T k,l (K) , with the hope of identifying it as such. Note that d ( θ t ) = e t ( X ∪ X ) − e t ( X (cid:48) ∪ X (cid:48) ) = t (cid:88) j =0 e t − j ( X ) e j ( X ) − t (cid:88) j =0 e t − j ( X (cid:48) ) e j ( X (cid:48) )= e t ( X ) − e t ( X (cid:48) ) + t (cid:88) j =1 e t − j ( X ) (cid:0) e j ( X ) − e j ( X (cid:48) ) (cid:1) + t (cid:88) j =1 (cid:0) e t − j ( X ) − e t − j ( X (cid:48) ) (cid:1) e j ( X (cid:48) )= e t ( X ) − e t ( X (cid:48) ) + t (cid:88) j =1 e t − j ( X ) d ( ξ j ) + t (cid:88) j =1 (cid:0) e t − j ( X ) − e t − j ( X (cid:48) ) (cid:1) e j ( X (cid:48) ) . (4-29)This suggests that we recursively define Θ t := θ t − t (cid:88) j =0 e t − j ( X ) ξ j − t (cid:88) j =0 Θ t − j e j ( X (cid:48) ) . (4-30)By (4-29), this gives d (Θ t ) = e t ( X ) − e t ( X (cid:48) ) . (4-31)and, in particular, d (Θ t ) = 0 for t > k .It then follows that we have quasi-isomorphisms T k,l (K) ∼ = a l q - l ( k + l + K (cid:2) e r ( X ) , e r ( X (cid:48) ) , e s ( X ) , e s ( X (cid:48) ) (cid:3) ⊗ K (cid:86) • { θ t , ξ s }∼ = a l q - l ( k + l + K (cid:2) e r ( X ) , e r ( X (cid:48) ) , e s ( X ) (cid:3) ⊗ K (cid:86) • { θ t }∼ = a l q - l ( k + l + R k ⊗ K K (cid:2) e s ( X ) (cid:3) ⊗ K (cid:86) • { Θ b } , (4-32)where, in this last equation, the index b ranges from k + 1 , . . . , k + l .This implies that T k,l (K) is quasi-isomorphic to a direct sum of a l q - l ( k + l + l (cid:89) i =1 1+ a - q k + i ) − q i = l (cid:89) i =1 q k + aq - k - i − q i (4-33)copies of R k , as desired. (cid:3) Lemma 4.11.
Let I = ( k , . . . , k r ) , J = ( k , . . . , k r − + k r ) , B ∈ D b (cid:0) R J - R I Bim q (cid:1) and C ∈ D b (cid:0) R I - R J Bim q (cid:1) , then we have B ...... k r k r − ∼ = q k r − k r B ...... k r − + k r C ...... k r k r − ∼ = q k r − k r C ...... k r − + k r (4-34) Proof.
We show the first quasi-isomorphism in (4-34), as the proof of the second is similar.The idea for the proof is easy: simply pass to a Koszul resolutions at the places where thetensor products take place. Formally, let us consider the case when r = 2 now, as the generalproof differs only in requiring more cumbersome notation. The left-hand side of the firstisomorphism in (4-34) is T I − (cid:0) T I ( I S J ⊗ R J B) (cid:1) ∼ = a k + k q - k - k - k - k (cid:0) I S J ⊗ R J J B I (cid:48) (cid:1) ⊗ K (cid:86) • { ξ r , ζ s } , (4-35)where ≤ r ≤ k , ≤ s ≤ k and with differential given by d ( ξ r ) = e r ( X ) − e r ( X (cid:48) ) , d ( ζ s ) = e s ( X ) − e s ( X (cid:48) ) for alphabets of size | X | = k = | X (cid:48) | and | X | = k = | X (cid:48) | ,respectively. Here, polynomials in the relevant alphabets act as indicated by the subscriptson the bimodules. Passing to a Koszul resolution of the diagonal R J -bimodule, we see this isquasi-isomorphic to a k + k q - k - k - k - k (cid:0) I S J ⊗ K J (cid:48) B I (cid:48) (cid:1) ⊗ K (cid:86) • { ξ r , ζ s , θ t } , (4-36)where here (additionally) ≤ t ≤ k + k , d ( θ t ) = e t ( X ) − e t ( X (cid:48) ) , and | X | = k + k = | X (cid:48) | .Similarly, the right-hand side is q k k T J (cid:0) B ⊗ R I I S J (cid:1) ∼ = a k + k q k k q - ( k + k )( k + k +1) (cid:0) J (cid:48) B I ⊗ R I I S J (cid:1) ⊗ K (cid:86) • { Θ t } , (4-37)where ≤ t ≤ k + k , d (Θ t ) = e t ( Y (cid:48) ) − e t ( Y ) , and | Y | = k + k = | Y (cid:48) | . Passing to a Koszulresolution of R I gives that this is quasi-isomorphic to a k + k q - ( k + k )( k + k +1)+2 k k (cid:0) J (cid:48) B I (cid:48) ⊗ K I S J (cid:1) ⊗ K (cid:86) • { Θ t , Ξ r , Z s } , (4-38)with ≤ r ≤ k , ≤ s ≤ k and differential given by d (Ξ r ) = e r ( Y (cid:48) ) − e r ( Y ) , d ( Z s ) = e s ( Y (cid:48) ) − e s ( Y ) for alphabets of size | Y | = k = | Y (cid:48) | and | Y | = k = | Y (cid:48) | . The result nowfollows by comparing (4-36) with (4-38). (cid:3) Lemma 4.12.
For k ≥ , there are atq -degree isomorphisms kk (cid:39) t k q - k kk & kk (cid:39) a k q - k - k kk (4-39)This result, which implies the invariance of the usual colored triply-graded link homologyunder stabilization, is well-known, and follows from the equivalence of the definition in termsof singular Soergel bimodules with the constructions in [WW17] and [Cau17]. We give the(well-known) argument for the sake of completion, and to determine the exact degree shifts(given our grading conventions for the Rickard–Rouquier complexes) so that we may be precisein (4-46) below. Proof.
We induct on k , starting with k = 1 . By Example 3.10 and (4-24), we have (cid:39) q + aq - − q −→ tq - + atq - − q & (cid:39) t - q + at - q - − q −→ q + aq - − q (4-40)The proof of [Hog18, Proposition 3.10] identifies the differentials in these complexes, givinghomotopy equivalences (cid:39) tq - & (cid:39) aq - (4-41) OMFLYPT HOMOLOGY FOR LINKS IN HANDLEBODIES 23 that follow from “Gaussian elimination” of all terms for which the aq -degrees coincide. Forthe inductive step, we compute, using Lemma 4.11 and (3-22), that q k − q - k q − q - k (cid:39) kk (cid:39) q - k - kk k − (cid:39) a k - q - k + k + kk (cid:39) a k - q - k + k + kk (cid:39) a k q - k + k - kk (cid:39) q k − q - k q − q - a k q - k - k kk (4-42)and the result follows for the negative crossing using the Krull–Schmidt property of thederived category, see e.g. [Wu14, Lemma 4.20]. The case of the positive crossing follows froman analogous computation. (cid:3) Together with (4-23), Lemma 4.12 proves stabilization invariance of HH • ℋ g ( 𝒷 ) (up tograding shift), and consequently completes the proof of Theorem 4.7.Hence, given a balanced, coloring ( 𝒷 , M ) of a handlebody braid, we define w ( 𝒷 , M ) := ∞ (cid:88) k =1 k · (cid:18) (cid:16) kk (cid:17) − (cid:16) k k (cid:17)(cid:19) , (4-43) i.e. it is a weighted sum of the difference between the number of purely k -colored positiveand negative crossings. Similarly, define W ( 𝒷 , M ) := ∞ (cid:88) k =1 k · (cid:18) (cid:16) kk (cid:17) − (cid:16) k k (cid:17)(cid:19) . (4-44)Passing to half-integral values of the at -gradings, we set x ( 𝒷 , M ) := a ( w ( 𝒷 , M ) − (cid:80) ni =1 l i ) t ( − w ( 𝒷 , M ) − (cid:80) ni =1 l i ) q − W ( 𝒷 , M ) + (cid:80) ni =1 l i + l i (4-45)and define HHH • ℋ g ( 𝒷 , M ) := H • ( x ( 𝒷 , M ) HH • ℋ g ( 𝒷 )) , (4-46)where H • ( − ) denotes taking homology. Corollary 4.13.
For balanced, colored ( 𝒷 , M ) ∈ ℬ r( g, n ) , the triply-graded vector space HHH • ℋ g ( 𝒷 , M ) ∈ K Vec atq is an invariant of the handlebody link 𝒷 ⊂ ℋ g . In general, HHH • ℋ g ( 𝒷 , M ) is not an invariant of the link corresponding to the closure in 𝒮 of thenon-core strands in 𝒷 . (cid:3) Proof.
First observe that our normalization factor (4-45) is invariant under the relations inthe handlebody braid group, i.e. if 𝒷 , 𝒷 (cid:48) ∈ ℬ r( g, n ) are colored handlebody braids related by(2-2), (2-3), (2-4), then x ( 𝒷 , M ) = x ( 𝒷 (cid:48) , M ) . Thus, since HH • ℋ g ( − ) is an invariant of handlebodybraids, the same is true for HHH •− ( 𝒷 , M ) . Conjugation invariance follows from the conjugationinvariance of HH • ℋ g ( 𝒷 ) , up to homotopy, given in Theorem 4.7, together with the observation that x ( 𝒷 , M ) = x ( 𝓈𝒷𝓈 − , 𝓈 · M ) (here 𝓈 · M is obtained from M by applying the permutationcorresponding to 𝓈 ).Invariance under stabilization follows from (4-23) and (4-39), together with a carefulinspection of (4-45).Finally, the second statement follows from (the proof of) Proposition 4.8, since this showsthat the homology of HH • ℋ g for the braids therein are not isomorphic up to a degree shift. (cid:3) References [All02] D. Allcock. Braid pictures for Artin groups.
Trans. Amer. Math. Soc. , 354(9):3455–3474, 2002.URL: https://arxiv.org/abs/math/9907194 , doi:10.1090/S0002-9947-02-02944-6 .[APS04] M.M. Asaeda, J.H. Przytycki, and A.S. Sikora. Categorification of the Kauffman bracket skeinmodule of I -bundles over surfaces. Algebr. Geom. Topol. , 4:1177–1210, 2004. URL: https://arxiv.org/abs/math/0409414 , doi:10.2140/agt.2004.4.1177 .[Bri73] E. Brieskorn. Sur les groupes de tresses [d’après V. I. Arnol’d] . Springer, Berlin, 1973.[Cau17] S. Cautis. Remarks on coloured triply graded link invariants.
Algebr. Geom. Topol. , 17(6):3811–3836,2017. URL: https://arxiv.org/abs/1611.09924 , doi:10.2140/agt.2017.17.3811 .[DR18] Z. Daugherty and A. Ram. Two boundary Hecke Algebras and combinatorics of type C. 2018. URL: https://arxiv.org/abs/1804.10296 .[El18] B. Elias. Gaitsgory’s central sheaves via the diagrammatic Hecke category. 2018. URL: https://arxiv.org/abs/1811.06188 .[EK10] B. Elias and M. Khovanov. Diagrammatics for Soergel categories. Int. J. Math. Math. Sci. , pagesArt. ID 978635, 58, 2010. URL: https://arxiv.org/abs/0902.4700 , doi:10.1155/2010/978635 .[EL17] B. Elias and I. Losev. Modular representation theory in type A via Soergel bimodules. 2017. URL: https://arxiv.org/abs/1701.00560 .[ESW17] B. Elias, N. Snyder, and G. Williamson. On cubes of Frobenius extensions. In Representationtheory—current trends and perspectives , EMS Ser. Congr. Rep., pages 171–186. Eur. Math. Soc.,Zürich, 2017. URL: https://arxiv.org/abs/1308.5994 .[EW14] B. Elias and G. Williamson. The Hodge theory of Soergel bimodules.
Ann. of Math. (2) , 180(3):1089–1136, 2014. URL: https://arxiv.org/abs/1212.0791 , doi:10.4007/annals.2014.180.3.6 .[GL97] M. Geck and S. Lambropoulou. Markov traces and knot invariants related to Iwahori–Heckealgebras of type B . J. Reine Angew. Math. , 482:191–213, 1997. URL: https://arxiv.org/abs/math/0405508 .[GGS18] E. Gorsky, S. Gukov, and M. Stošić. Quadruply-graded colored homology of knots.
Fund. Math. ,243(3):209–299, 2018. URL: https://arxiv.org/abs/1304.3481 , doi:10.4064/fm30-11-2017 .[GNR16] E. Gorsky, A. Negut, and J. Rasmussen. Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology 2016. URL: https://arxiv.org/abs/1608.07308 .[GLW18] J.E. Grigsby, A.M. Licata, and S.M. Wehrli. Annular Khovanov homology and knotted Schur–Weylrepresentations. Compos. Math. , 154(3):459–502, 2018. URL: https://arxiv.org/abs/1505.04386 , doi:10.1112/S0010437X17007540 .[GW10] J.E. Grigsby and S.M. Wehrli. Khovanov homology, sutured Floer homology, and annular links. Algebr. Geom. Topol. , 10:2009–2039, 2010. URL: https://arxiv.org/abs/0907.4375 , doi:10.2140/agt.2010.10.2009 .[GS12] S. Gukov and M. Stošić. Homological algebra of knots and BPS states. In Proceedings of theFreedman Fest , volume 18 of
Geom. Topol. Monogr. , pages 309–367. Geom. Topol. Publ., Coventry,2012. URL: https://arxiv.org/abs/1112.0030 , doi:10.2140/gtm.2012.18.309 .[HOL02] R. Häring-Oldenburg and S. Lambropoulou. Knot theory in handlebodies. J. Knot Theory Ramifi-cations , 11(6):921–943, 2002. Knots 2000 Korea, Vol. 3 (Yongpyong). URL: https://arxiv.org/abs/math/0405502 , doi:10.1142/S0218216502002050 .[Hog18] M. Hogancamp. Categorified Young symmetrizers and stable homology of torus links. Geom.Topol. , 22(5):2943–3002, 2018. URL: https://arxiv.org/abs/1505.08148 , doi:10.2140/gt.2018.22.2943 .[ILZ18] K. Iohara, G. Lehrer, and R. Zhang. Schur–Weyl duality for certain infinite dimensional U q ( sl ) -modules. 2018. URL: https://arxiv.org/abs/1811.01325 .[Jon87] V.F.R. Jones. Hecke algebra representations of braid groups and link polynomials. Ann. of Math.(2) , 126(2):335–388, 1987. doi:10.2307/1971403 .[Kho00] M. Khovanov. A categorification of the Jones polynomial.
Duke Math. J. , 101(3):359–426, 2000.URL: http://arxiv.org/abs/math/9908171 , doi:10.1215/S0012-7094-00-10131-7 . OMFLYPT HOMOLOGY FOR LINKS IN HANDLEBODIES 25 [Kho07] M. Khovanov. Triply-graded link homology and Hochschild homology of Soergel bimodules.
Internat. J. Math. , 18(8):869–885, 2007. URL: https://arxiv.org/abs/math/0510265 , doi:10.1142/S0129167X07004400 .[KR08a] M. Khovanov and L. Rozansky. Matrix factorizations and link homology. Fund. Math. , 199(1):1–91,2008. URL: https://arxiv.org/abs/math/0401268 , doi:10.4064/fm199-1-1 .[KR08b] M. Khovanov and L. Rozansky. Matrix factorizations and link homology. II. Geom. Topol. , 12(3):1387–1425, 2008. URL: https://arxiv.org/abs/math/0505056 , doi:10.2140/gt.2008.12.1387 .[Lam93] S. Lambropoulou. A study of braids in -manifolds. 1993. Ph.D. thesis, University of Warwick.URL: http://wrap.warwick.ac.uk/73390/ .[Lam00] S. Lambropoulou. Braid structures in knot complements, handlebodies and -manifolds. In Knots inHellas ’98 (Delphi) , volume 24 of
Ser. Knots Everything , pages 274–289. World Sci. Publ., River Edge,NJ, 2000. URL: https://arxiv.org/abs/math/0008235 , doi:10.1142/9789812792679_0017 .[MSV11] M. Mackaay, M. Stošić, and P. Vaz. The , -coloured HOMFLY–PT link homology. Trans. Amer.Math. Soc. , 363(4):2091–2124, 2011. URL: https://arxiv.org/abs/0809.0193 , doi:10.1090/S0002-9947-2010-05155-4 .[OR07] R. Orellana and A. Ram. Affine braids, Markov traces and the category O . Algebraic groups andhomogeneous spaces , 423–473, Tata Inst. Fund. Res. Stud. Math., 19, Tata Inst. Fund. Res., Mumbai,2007. URL: https://arxiv.org/abs/math/0401317 .[QR18] H. Queffelec and D.E.V. Rose. Sutured annular Khovanov–Rozansky homology.
Trans. Amer.Math. Soc. , 370(2):1285–1319, 2018. URL: https://arxiv.org/abs/1506.08188 , doi:10.1090/tran/7117 .[QRS18] H. Queffelec, D.E.V. Rose, and A. Sartori. Annular evaluation and link homology. 2018. URL: https://arxiv.org/abs/1802.04131 .[Rob13] L.P. Roberts. On knot Floer homology in double branched covers. Geom. Topol. , 17:413–467, 2013.URL: https://arxiv.org/abs/0706.0741 , doi:10.2140/gt.2013.17.413 .[RT] D.E.V. Rose and D. Tubbenhauer. HOMFLYPT homology for links in handlebodies via type C Soergel bimodules. In preparation.[Rou06] R. Rouquier. Categorification of sl and braid groups. In Trends in representation theory of algebrasand related topics , volume 406 of
Contemp. Math. , pages 137–167. Amer. Math. Soc., Providence,RI, 2006. URL: https://arxiv.org/abs/math/0409593 , doi:10.1090/conm/406/07657 .[Rou17] R. Rouquier. Khovanov–Rozansky homology and -braid groups. In Categorification in geometry,topology, and physics , volume 684 of
Contemp. Math. , pages 147–157. Amer. Math. Soc., Providence,RI, 2017. URL: https://arxiv.org/abs/1203.5065 .[SW18] N. Saunders and A. Wilbert. Exotic Springer fibers for orbits corresponding to one-row bipartitions.2018. URL: https://arxiv.org/abs/1810.03731 .[Soe92] W. Soergel. The combinatorics of Harish-Chandra bimodules.
J. Reine Angew. Math. , 429:49–74,1992. doi:10.1515/crll.1992.429.49 .[TVW17] D. Tubbenhauer, P. Vaz, and P. Wedrich. Super q -Howe duality and web categories. Algebr.Geom. Topol. , 17(6):3703–3749, 2017. URL: https://arxiv.org/abs/1504.05069 , doi:10.2140/agt.2017.17.3703 .[Ver98] V.V. Vershinin. On braid groups in handlebodies. Sibirsk. Mat. Zh. , 39(4):755–764, i, 1998. doi:10.1007/BF02673050 .[WW11] B. Webster and G. Williamson. The geometry of Markov traces.
Duke Math. J. , 160(2):401–419,2011. URL: https://arxiv.org/abs/0911.4494 , doi:10.1215/00127094-1444268 .[WW17] B. Webster and G. Williamson. A geometric construction of colored HOMFLYPT homology. Geom.Topol. , 21(5):2557–2600, 2017. URL: https://arxiv.org/abs/0905.0486 , doi:10.2140/gt.2017.21.2557 .[Wil11] G. Williamson. Singular Soergel bimodules. Int. Math. Res. Not. IMRN , (20):4555–4632, 2011.URL: https://arxiv.org/abs/1010.1283 , doi:10.1093/imrn/rnq263 .[Wu14] H. Wu. A colored sl ( N ) homology for links in S . Dissertationes Math. , 499:217, 2014. URL: https://arxiv.org/abs/0907.0695 , doi:10.4064/dm499-0-1 . D.E.V.R.: Department of Mathematics, North Carolina at Chapel Hill, Phillips HallCB3250, UNC-CH, Chapel Hill, NC 27599-3250, United states, davidev.web.unc.edu
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