ABSTRACT. We consider the construction of refined Chern-Simons torus knot invariants by M. Aganagic and S. Shakirov from the DAHA viewpoint of I. Cherednik. We give a proof of Cherednik's conjecture on the stabilization of superpolynomials, and then use the results of O. Schiffmann and E. Vasserot to relate knot invariants to the Hilbert scheme of points on C 2 . Then we use the methods of the second author to compute these invariants explicitly in the uncolored case. We also propose a conjecture relating these constructions to the rational Cherednik algebra, as in the work of the first author, A. Oblomkov, J. Rasmussen and V. Shende. Among the combinatorial consequences of this work is a statement of the m n shuffle conjecture.

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Recent work of the first author, Negut , and Rasmussen, and of Oblomkov and Rozansky in the context of Khovanov-Rozansky knot homology produces a family of polynomials in q and t labeled by integer sequences. These polynomials can be expressed as equivariant Euler characteristics of certain line bundles on flag Hilbert schemes. The q, t-Catalan numbers and their rational analogues are special cases of this construction. In this paper, we give a purely combinatorial treatment of these polynomials and show that in many cases they have nonnegative integer coefficients.For sequences of length at most 4, we prove that these coefficients enumerate subdiagrams in a certain fixed Young diagram and give an explicit symmetric chain decomposition of the set of such diagrams. This strengthens results of Lee, Li and Loehr for (4, n) rational q, t-Catalan numbers.

ABSTRACT. We conjecturally extract the triply graded Khovanov-Rozansky homology of the (m, n) torus knot from the unique finite dimensional simple representation of the rational DAHA of type A, rank n − 1, and central character m/n. The conjectural differentials of Gukov, Dunfield and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the doubly graded Khovanov-Rozansky homologies. We match our conjecture to previous conjectures of the first author relating knot homology to q, t-Catalan numbers, and of the last three authors relating knot homology to Hilbert schemes on singular curves.

We conjecture the existence of four independent gradings in the colored HOMFLY homology. We describe these gradings explicitly for the rectangular colored homology of torus knots and make qualitative predictions of various interesting structures and symmetries in the colored homology of general knots. We also give a simple representation-theoretic model for the HOMFLY homology of the unknot colored by any representation. While some of these structures have a natural interpretation in the physical realization of knot homologies based on counting supersymmetric configurations (BPS states, instantons, and vortices), others are completely new. They suggest new geometric and physical realizations of colored HOMFLY homology as the Hochschild homology of the category of branes in a Landau-Ginzburg B-model or, equivalently, in the mirror A-model. Supergroups and supermanifolds are surprisingly ubiquitous in all aspects of this work.

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We introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund's bijection ζ exchanging the pairs of statistics (area, dinv) and (bounce, area) on Dyck paths, and the Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constructions. We also relate our combinatorial constructions to representation theory. We derive new formulas for the Poincaré polynomials of certain affine Springer fibers and describe a connection to the theory of finite dimensional representations of DAHA and nonsymmetric Macdonald polynomials. Proposition 1.1. If m and n are coprime then m-stable affine permutations label the alcoves in a certain simplex D m n which is isometric to the m-dilated fundamental alcove. In particular, the number of m-stable affine permutations equals m n−1 .The simplex D m n (first defined in [10,27]) plays the central role in our study. We show that the alcoves in it naturally label various algebraic and geometric objects such as cells in certain affine Springer fibres and nonsymmetric Macdonald polynomials at q m = t n . We provide a clear combinatorial dictionary that allows one to pass from one description to another.We define two maps A, PS between the m-stable affine permutations and m n-parking functions and prove the following results about them. Theorem 1.2. Maps A and PS satisfy the following properties:(1) The map A is a bijection for all m and n.(2) The map PS is a bijection for m = kn ± 1. For m = kn + 1, it is equivalent to the Pak-Stanley labeling of Shi regions. 1 2 EUGENE GORSKY, MIKHAIL MAZIN, AND MONICA VAZIRANI (3) The map PS ○ A −1 generalizes the bijection ζ constructed by Haglund in [18]. More concretely, if one takes m = n + 1 and restricts the maps A and PS to minimal length right coset representatives of S n S n , then PS ○ A −1 specializes to Haglund's ζ. Remark 1.3. For m = n + 1 the bijection A is similar to the Athanasiadis-Linusson [5] labeling of Shi regions, but actually differs from it. Conjecture 1.4. The map PS is bijective for all relatively prime m and n.The map PS has an important geometric meaning. In [24] Lusztig and Smelt considered a certain Springer fibre F m n in the affine flag variety and proved that it can be paved by m n−1 affine cells. In [14,15] a related subvariety of the affine Grassmannian has been studied under the name of Jacobi factor, and a bijection between its cells and the Dyck paths in m × n rectangle has been constructed. In [20] Hikita generalized this combinatorial analysis and constructed a bijection between the cells in the affine Springer fiber and m n-parking functions (in slightly different terminology). He gave a quite involved combinatorial formula for the dimension of ...

We compute the Heegaard Floer link homology of algebraic links in terms of the multivariate Hilbert function of the corresponding plane curve singularities. The main result of the paper identifies four homologies: (a) the Heegaard Floer link homology of the local embedded link, (b) the lattice homology associated with the Hilbert function, (c) the homologies of the projectivized complements of local hyperplane arrangements cut out from the local algebra, and (d) a generalized version of the Orlik-Solomon algebra of these local arrangements. In particular, the Poincaré polynomials of all these homology groups are the same, and we also show that they agree with the coefficients of the motivic Poincaré series of the singularity. 10-01-678, RFBR-13-01-00755, NSh-8462.2010.1, NSF grant DMS-1403560 and the Simons foundation. A. N. is partially supported by OTKA Grants 81203, 100796 and 112735. HEEGAARD FLOER LINK HOMOLOGY2.1. Review of Heegaard Floer link homology. In this subsection we recall certain basic algebraic structures of Heegaard Floer link homology. For more see [15,27,28,29,30,33].