Research Statement Michael Abel October 2017

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1 Categorification can be thought of as the process of adding an extra level of categorical structure to a well-known algebraic structure or invariant. Possibly the most famous example (ahistorically speaking) of this process is the singular homology functor H. In particular, if we take the alternating sum of the dimensions of H i (X) we recover the Euler characteristic, a well-understood invariant of topological spaces. Not only is the isomorphism type of H (X) a stronger invariant of the topological space X, but H is a functor. Furthermore, H (f) is also as an invariant of maps up to homotopy. My research primarily focuses on studying various structures, coming from the representation theory of U q (sl n ), on the sl N and HOMFLY-PT link homology theories of Khovanov and Rozansky and the related topology. These theories categorify the corresponding knot polynomial, that is for a link L the graded Euler characteristic of H sl N (L) is equal to P sl N (L) and similarly for the HOMFLY-PT polynomial. In this research statement I will describe the following motivating goals and specific projects along two separate, but related, directions: The image of the infinite full twist braid in link homology was first studied by Rozansky for Khovanov homology in [41]. He showed that these objects categorified the highest weight projectors for U q (sl 2 ). Hogancamp showed this approach works in the categorification of Hecke algebras, the homotopy category of Soergel bimodules [21]. The categorified highest weight projectors can be used to construct categorifications of colored link homology theories and study stabilization of the link homology of (n, k) torus links as k. Hogancamp and I in [3] construct a categorification of the lowest weight projectors, or onecolumn Young symmetrizers, for the Hecke algebra. The Hochschild homology of these objects, HH(P 1 n), correspond to limits of HOMFLY-PT homology of (n, nk + m) torus links as k. We are able to use a combinatorial construction of P 1 n to explicitly compute these limiting homologies. Twisted versions of P 1 n can be used to construct non-isomorphic categorifications of the Λ n -colored HOMFLY-PT polynomial. In upcoming work with Hogancamp [2], we give computations in specific cases including for torus links in colored HOMFLY-PT homology. In particular, we present a recursive combinatorial algorithm to compute the Sym n -colored HOMFLY-PT homology for any torus link. With this algorithm we give rigorous evidence for many conjectures for physics and introduce new conjectures of our own. In recent work with Willis [6] we show that the limiting complex for any infinite, positive, complete braid will be isomorphic to the a highest weight projector in colored sl N and HOMFLY-PT homology. Different limits (such as the lowest weight limit mentioned above) are expected to have similar properties, given the appropriate restrictions One construction of HOMFLY-PT homology uses complexes of Soergel bimodules and Hochschild homology [25, 39]. Soergel bimodules have many different filtrations by standard bimodules or virtual crossings [4, 42, 44]. With Rozansky in [4] I explore these virtual filtrations in HOMFLY-PT homology. We proved that they give an invariant under Reidemeister I and II and we gave a bound on the possible error given by Reidemeister III. Using virtual filtrations in [1], I show that HOMFLY- PT homology is not invariant under the oriented Reidemeister IIb move. This was expected from the original construction [28], but had yet to be explicitly confirmed. In recent years, annular versions of Khovanov and Khovanov-Rozansky sl N homology have been studied by various groups [7, 35]. We can explore if a similar construction exists for HOMFLY-PT homology. However, the failure of Reidemeister IIb complicates the construction. Using the techniques from [1], Wedrich and I are studying a construction of an anuular HOMFLY-PT homology theory, while trying to take into account the obstructions posed by the failure of Reidemeister IIb. Virtual filtrations can also arise in sl N homology [26, 27]. Conjecturally these filtrations are also invariant under the Reidemeister moves. In work in preparation with Dmitry Vagner [5], we explore a smooth dotted TQFT construction of sl N homology using virtual filtrations to resolve MOY graphs into collections of disjoint circles. 1

2 Before describing these projects in more detail, I will first give some historical context and background material on Khovanov and Rozansky s HOMFLY-PT homology. 1 Categorification and Khovanov-Rozansky Homology The term categorification, first introduced by Crane and Frenkel [13], refers to a process in which we construct a category C whose Groethendieck group K 0 (C) is the algebraic structure A we wished to categorify. In particular, for homological link invariants we wish to find an object X in a new category C whose image in K 0 (C) is the orginial link invariant. The first example of this process for link invariants arose in Khovanov s seminal paper defining what is now known as Khovanov homology [24]. Khovanov homology categorified the Jones polynomial by lifting the Kauffman bracket relation to a short exact sequence of chain complexes of graded Z-modules. This construction is relatively computable and accessible [8, 9], and it has been shown to be a stronger invariant than the Jones polynomial [8], functorial with respect to link cobordisms [11], and an unknot detector [30]. This additional structure given by categorification has been used to give topological applications such as Rasmussen s combinatorial proof of the Milnor conjecture [37]. This was originally proven by Kronheimer and Mrowka using gauge theory [29]. In 2004 Khovanov and Rozansky introduced a categorification of the sl N link invariant [27]. This was specifically a categorification of the MOY construction of the sl N polynomial using directed weighted graphs [33]. Khovanov-Rozansky sl N homology is also functorial with respect to link cobordisms [15, 36]. In 2006 Khovanov and Rozansky introduced a homology theory which categorfied the entire HOMFLY-PT polynomial as a triply-graded homology theory [28]. This theory must be constructed from the closure of a braid rather than a link diagram [1, 28]. Khovanov later showed that HOMFLY-PT homology was related to Rouquier s map of the braid group to the homotopy category of Soergel bimodules [25, 39]. I will now give a quick description of HOMFLY-PT homology. Let x and y denote lists of variables x 1,..., x n and y 1,..., y n respectively. Let A denote the category of Q[x]-Q[y]-bimodules (or equivalently the category of Q[x, y]-modules) and let K(A) denote its homotopy category of chain complexes. We now define two basic Soergel bimodules via diagrammatics; = Q[x 1, x 2, y 1, y 2 ]/(x 1 y 1, x 2 y 2 ), = Q[x 1, x 2, y 1, y 2 ]/(x 1 + x 2 y 1 y 2, x 1 x 2 y 1 y 2 ). Rouquier gave a map which sends braids in the braid group Br n to complexes of bimodules in K(A). Rouquier s map, which we will denote by F, sends composition of braids to composition of bimodules. In particular, F(α β) = F(α) F(β). F is defined by ( ) F = χ +, F ( ) = χ, where χ + and χ are canonical inclusion and projection maps. Rouquier in [39] proved that these complexes give an action of the braid group on K(A). Khovanov in [25] showed that this action can be used to construct HOMFLY-PT homology. Define HH to be the functor on K(A) which applies Hochschild homology to each chain module and differential. Then, in particular Khovanov proved that H(β) = H(HH(F(β))) is a link invariant isomorphic to the HOMFLY-PT homology of [28]. Note that H(β) does not depend on the braid presentation of L and is indeed a link invariant. 2 Virtual crossings and filtrations in link homology We now introduce a new bimodule which is defined by = Q[x 1, x 2, y 1, y 2 ]/(x 1 y 2, y 2 x 1 ). We can think of this bimodule as the result of permuting the variables y 1 and y 2 in. We will call a virtual crossing as they are used in constructing an action of the virtual braid group on the category of Q[x]-Q[y]- bimodules in work of Thiel [44]. There are two filtrations of in terms of virtual crossings. These filtrations have been studied to understand the structure of the category of Soergel bimodules, denoted by SBim n, and are part of an algebraic proof of the Kazdahn-Lusztig conjectures in the A-type case [42]. We have two short exact sequences 0 χ + χ 0, and

3 The first short exact sequence describes a filtration on, where. We will denote with this filtration by +. Likewise the second short exact sequence describes a filtration. We will denote with this filtration by. Note that χ + and χ both are filtered maps. We now define a filtered version of the Rouquier functor, which we will call F V, by ( ) ( ) χ + F V = + F V = χ. For a general braid β Br n, we give F V (β) the tensor product filtration. Note that the filtrations on chain bimodules arising in F V (β) may be different than those studied in [42]. H(β) inherits the filtration of F V (β). We will use the notation H V to specify that we wish to consider H(β) as a filtered vector space. The filtration on F V (β) and H V (β) will be referred to as the virtual filtration in the sequel. Recall, two braid words are Markov equivalent if one can be transformed into the other via a sequence of conjugations and stabilizations, that is interchanging a word β Br n with βσ n ±1 Br n+1. Theorem 2.1 (A.-Rozansky [4]). If α and β are equivalent braid words (that is, not related via the braid relation) in Br n, then F V (α) is filtered homotopy equivalent to F V (β). Suppose two braid words α and β (not necessarily in the same number of strands) are Markov equivalent, then H V (α) is filtered isomorphic as a triply-graded vector space to H V (β). Conjecture 2.2. If α and β differ by the braid relation, then H V (α) is filtered isomorphic as a triply-graded vector space to H V (β). I have the following result in the direction of proving Conjecture 2.2. Theorem 2.3 (A.-Rozansky [4]). Suppose β and β are two braids related by the braid relation, then the (possibly nonfiltered) isomorphism between H V (β) and H V (β ) violates filtration by at most two levels. The definition of F (and F V ) can be expanded to arbitrary (oriented) virtual tangles by working in the homotopy category of the derived category of bimodules (See [1, 28] for more details). It was expected in [28] that it was necessary to use a braid closure to construct HOMFLY-PT homology. The reason for this expectation was the difficulty in proving the oppositely oriented Reidemeister II move. In [1] I use virtual filtrations to give an explicit answer to this question. Theorem 2.4 (A. [1]). F( ) F( ). In particular, F( ) F( ), thus F( ) is not an isotopy invariant of tangles. Corollary 2.5 (A. [1]). The virtual exchange move (see [23]) fails. Thus, F( ) is not an invariant of virtual links. 2.1 Morphisms between unlinks in HOMFLY-PT homology Recent work by Naisse and Vaz [34], describes a diagrammatic calculus for computing HOMFLY-PT homology related to a certain parabolic 2-Verma module of gl 2n. Via my recent work [1], there should be a direct route of constructing a similar diagrammatic framework directly from the behavior of the derived bimodule construction of HOMFLYPT homology. In joint work with Wedrich, we explore this diagrammatic construction further in the hope of understanding an annular HOMFLY-PT homology theory. In particular, as expected from Naisse-Vaz, there are no degree 0 morphisms permuting two oppositely-oriented circles which factors through Reidemeister IIb morphisms. However, there are interesting extensions which do factor through the Reidemeister IIb morphisms. Using the computations from [1] we can explicitly compute these extensions. 2.2 A Smooth TQFT construction of Khovanov-Rozansky sl N homology In [26] Khovanov and Rozansky introduce an approach to constructing sl N homology using virtual crossings. To each of the diagrams, and they associate certain matrix factorizations. They prove that there exist maps of matrix factorizations F and G such that Cone(F : ) and Cone(G : ) are homotopy equivalent (as matrix factorizations) to. This allows us to resolve any MOY graph arising in sl N homology to a convolution of virtual crossings similar to what was described for the HOMFLY-PT case. 3

4 Using the natural filtrations of the convolutions, we may define a filtration for sl N homology. Analogues of Theorem 2.1, Conjecture 2.2 and Theorem 2.3 hold are expected to hold for sl N homology as well. As alluded to in [26], sl N homology is also an invariant of virtual links. Thus after resolving a virtual link into closed MOY graphs and then resolving the closed MOY graph into diagrams with only virtual crossings, we see that every term of our now larger complex is a unlink. The sl N homology of an unlink is A n, where A is the Frobenius algebra Q[x]/(x N ). After closing the diagrams and, the maps F and G are homotopic to the multiplication and comultiplication of A [26]. A natural question is if one can construct a smooth TQFT construction à la Khovanov s original construction of Khovanov homology [24]. In upcoming work with Dmitry Vagner, a graduate student at Duke, we write out this approach in detail. The outcome is a bicomplex with two commuting differentials d h and d v. d h is the graded differential from the original complex of matrix factorizations defining sl N homology. The components of d h are natural inclusions and projections of mapping cones. d v is a filtered differential coming from the differential on the convolutions of virtual crossings. Denote this bicomplex for a link diagram D by C N (D). Theorem 2.6 (A.-Vagner [5]). Suppose D is a diagram for a virtual link with n components. There exists a spectral sequence C(D) A n whose E 1 -page, with differential d h, is homotopy equivalent to the Khovanov-Rozansky complex for D and the E 2 -page is Khovanov-Rozansky sl N homology. We can view the filtered differential d v in the following way. The length 1 components (that move one step down in the filtration) will be described in terms of multiplication and comultiplication on A. Unfortunately the other components elude a combinatorial description. However, in [5] we give an explicit algorithm to construct these components via the data in the construction of the virtual filtration. An understanding of this construction could give new computational methods for sl N homology, new insight into topological invariants (such as concordance invariants from sl N homology [31]), and lifting constructions which rely on the TQFT construction of Khovanov homology to Khovanov-Rozansky homology. 3 Infinite torus braids and colored link homology In [43], Stošić proved that the Khovanov homology of (m, n)-torus knots approach a stable limit as n. This limit was later reinterpreted by Rozansky [41] by showing the image of the (m, km)-torus braid in Bar-Natan s category [9] approaches a stable limit as k. The stable limit of Rozansky categorifies a Jones-Wenzl projector which is the sl 2 specialization of the Young symmetrizer p (n). This work was extended to sl 3 by Rose [38] and sl N by Cautis [10]. Hogancamp in [21] extends this to the HOMFLY- PT case by categorifying the Young symmetrizer p (n) itself. We denote this object by P (n). By computing the Hochschild homology of P (n), Hogancamp computes the stable Khovanov-Rozansky homology of the (n, m)-torus link as m, proving a conjecture by Dunfield, Gukov, and Rasmussen [14, 21]. It was proven by Rozansky [40] that a second type of stable limit exists in Khovanov homology which corresponds to stabilizing the other end of the complex. This stable limit is also explicitly related to a recent conjecture of Gorsky, Negut and Rasmussen [20]. They predict that the structure of the Hochschild (co)homology of categorified Young symmetrizers in general can be determined by the local algebras of torus fixed points in flag Hilbert scheme of n points on C 2. We let R = Q[x 1,..., x n ] and B = R R Sn R. B, with 1 1 shifted to be in degree n(1 n)/2, is the Soergel bimodule associated to the longest word of S n. Let p 1 n denote the Young antisymmetrizer in the Hecke algebra of S n. Hogancamp and I construct a categorical analogue of this object in the homotopy category of Soergel bimodules, K(SBim n ). Let FT k denote the image of the (n, nk) torus braid in K(SBim n ). Theorem 3.1 (A.-Hogancamp [3]). There exists a bounded from below complex P 1 n in K(SBim n ) and a map ε : P 1 n R such that the following conditions hold. 1. Every chain bimodule of P 1 n is a direct sum of shifted copies of B. 2. B R Cone(ε) Cone(ε) R B 0 3. The image of P 1 n in the Groethendieck group K 0 (K(SBim n )), equivalently the Hecke algebra of S n, is equal to p 1 n. 4

5 4. P 1 n is a resolution of R by free B-modules. 5. There exist maps f k : FT k 1 FT k such that P 1 n is homotopy equivalent to holim (FT k, f k ). Any complex P that satisfies conditions (1)-(3) is (canonically) homotopy equivalent to P 1 n. We will call a bounded from below complex P 1 n as above a categorified Young antisymmetrizer. There also exists a categorical S n -action on P 1 n. The differential on P 1 n is given by matrices with polynomial entries in Q[x, y]. Let w S n and let w(p 1 n) denote P 1 n with w permuting the indices of the y variables of polynomials describing the maps.w(p 1 n) is a resolution of R w by free B-modules, where R w is the bimodule R whose right action is twisted by w. We will denote deg(z) = (i, j, k) instead as a monomial deg(z) = q i a j t k. We prove in [3] that P 1 n admits a nice combinatorial description as a dg-module. Definition 3.2. Suppose x = y = n with deg(x i ) = deg(y i ) = q 2. Let u = u 1,..., u n be even variables of degree deg(u i ) = t 2 q 2i, and θ = θ 1,..., θ n be odd variables of degree deg(θ i ) = q 2 t. Define the dg-q[x, y]- algebra A n = Q[x, y, u, θ] with Q[x, y]-linear differential d A (u k ) = 0, d A (θ k ) = k a ik (x, y) for special polynomials a ik which can be described in terms of double Schubert polynomials. Also let I n = (e 1 (x) e 1 (y),..., e n (x) e n (y)). Then we define the dg-a n -module M n = A n /I n shifted so that deg(1) = q n(n 1)/2 with the unique differential d M such that d M (1) = n i=1 (x i y i )θ i. Theorem 3.3 (A.-Hogancamp [3]). M n, as an object of K(SBim n ), is isomorphic to P 1 n. If we permute y by w in the definitions of d A and d M, then we get an object which is isomorphic to w(p1 n). Using this combinatorial description, we explicitly compute H(w(P 1 n)) for all n and w. The following theorem was first conjectured by Gorsky, Negut and Rasmussen [20]. Theorem 3.4 (A.-Hogancamp [3]). For each 1 i < j n, let v ij denote an even variable of degree deg(v ij ) = q 2i 2j 2 t 2. Let X denote the n n matrix with x 1,..., x n on the diagonal and 1 s on the superdiagonal. Let V denote the n n matrix [v ij ] with v ij = 0 if i j. Let E denote the ring E = R[v ij ]/J where J is the ideal generated by the entries of the commutator [X, V ]. Then up to an overall grading shift, H(w(P 1 n)) H 0(w(P 1 n)) Q Λ[ξ 2,..., ξ n ], where H 0 (w(p 1 n)) is isomorphic to the quotient of E in which we identify x w(i) with x i for all 1 i n, and ξ 2,..., ξ n are odd variables with degree deg ξ m = a 2 q 2m. Corollary 3.5 (A.-Hogancamp [3]). The stable Khovanov-Rozansky homology of (n, nk + m) torus links is isomorphic to H(γ m n (P 1 n)), where γ n is an n-cycle in S n. 3.1 Computations in colored link homology In upcoming work [2], Hogancamp and I use w(p 1 n) to construct colored link homology theories. Theorem 3.6 (A.-Hogancamp [2]). Let β be a braid representative of a link L with k components and let w S n. If π is the underlying permutation of β, then let β n,w denote the n-parallel of β where one strand per cycle of π is cabled by w(p 1 n) and the rest are cabled by P 1 n. Then H(L, n, w) := H(HH(β n,w )) is a link invariant categorifying the 1 n -colored HOMFLY-PT polynomial. We expect that the homology of links colored by γ n (P 1 n) will factor into a free superpolynomial ring tensored with a reduced colored HOMFLY-PT homology. We expect that this reduced homology should be isomorphic to the (Webster-Williamson) colored HOMFLY-PT homology of the link. Using work of Elias-Hogancamp [17] and Mellit [32] to compute the Sym n -colored HOMFLY-PT homology of torus links. We give a surprisingly simple recursive algorithm for computing this homology (e.g; the Sym 3 -colored homology of the (4, 5)-torus link takes around 8 seconds for a common laptop). In the process, we show that the Sym n -colored HOMFLY-PT homology of a torus knot factors as a superpolynomial ring, H(P n ), and a finite-dimensional vector space. We call this finite-dimensional vector space the reduced Sym n -colored HOMFLY-PT homology. i=1 5

6 Using this algorithm, we give experimental evidence for exponential growth of the rank of colored HOMFLY-PT homology of torus links and compare our calculations with those expected from physics (See [19, 18]). In all cases checked thus far, our rigorous calculations agree with their conjectures (up to at most a regrading). 3.2 Khovanov-Rozansky Homology of infinite braids Willis and I in recent work look at the HOMFLY-PT homology of other infinite braids. In [22], Islambouli and Willis prove that the Khovanov complex of any positive, complete (all braid generators appear an infinite number of times), semi-infinite braid is homotopy equivalent to the Khovanov complex of the infinite full twist or the categorified Jones-Wenzl projector (See: [12, 41]). This result lifts to colored sl N and HOMFLY-PT homology. Let B be any positive, complete, semi-infinite braid viewed as the limit of braid words B = lim l σ j1 σ jl = lim l B l. Let B (m) denote the semi-infinite braid B colored by m (representing the mth fundamental representation) and similarly for B l,(m). The main result of our work is the following theorem. Theorem 3.7 (A.-Willis [6]). The limiting Khovanov-Rozansky sl(n) complex C N (B (m) ) = lim C N (B l,(m) ) exists for all m {1,..., N 1}. Furthermore, C N (B (m) ) C N (FT (m)). Likewise, the Khovanov-Rozansky HOMFLY-PT complex F(B (m) ) = lim F(B l,(m) ) exists for all m N and F(B (m) ) F(FT (m)). Recall a link is said to be braid positive if it is the closure of a positive braid β. Using the proof of Theorem 3.7, we prove the following partial isomorphism on the HOMFLY-PT homology of braid positive links. Theorem 3.8 (A.-Willis [6]). Let β be a finite positive braid on n strands with l crossings and suppose y(l) is the number of diagonals 1 in β. Then there exists a map F l : H(β) H(Tn y(l) ) which is an isomorphism for all homological degrees less than y(l). Willis and I expect that similar results should hold in other theories as well. Using the categorified Young symmetrizers P T, where T is a standard Young tableau, of Elias and Hogancamp [16], we expect that Theorem 3.7 will hold for colored HOMFLY-PT homology where we color by an arbitrary Young tableau T. Also, by adding additional restrictions to the inverse system constructing B, we expect to be able to write a similar result for multicolored braids. In particular, we want to (if possible) choose a subsequence of finite braids so that each braid fixes the labels given by the colors. Conjecture 3.9. Let T = {T 1,..., T n } be an ordered list of standard Young tableaux and B be a semi-infinite, positive, complete braid. Suppose that there exists a subsequence of finite subbraids of B, {B lk } k N such that the underlying permutation of B l fixes T. Let F T (B lk ) denote the Rouquier complex of the braid B lk cabled by {P T1,..., P Tn }. Then the inverse system {F(B lk ), f k }, where f k : F(B lk ) F(B lk 1 ) is a canonical projection map, has an inverse limit F(B ). We also expect this to work with other limits as well, such as Rozansky s lowest weight limit [40]. For example consider the categorified Young antisymmetrizer defined in Theorem 3.1. This is constructed as the Rouquier complex of an infinite braid, but shifted where the head stabilizes rather than the tail. We expect that this limit can be constructed in a similar manner to our result in Theorem 3.7. Conjecture Let B be a semi-infinite, positive, complete, periodic braid. For any subsequence of complete pure braids B lk, lim k ( )F(B lk ) P 1 n. Here ( ) denotes a grading shift depending on l k. A similar result should hold for sl N homology as well. If this result holds, then combining the result with Theorem 3.4 will give a partial isomorphism similar to the one in Theorem Roughly speaking, we define y(l) as the largest integer such that (σ 1 σ n 1 ) y(l) is a subbraid of β. See [6] for more details. 6

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