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Tautological classes for (n,n+1) torus knots

Eugene Gorsky, Anton Mellit

Abstract

We construct an explicit isomorphism between the HOMFLY-PT homology of $(n,n+1)$ torus knots and the direct sum of hook isotypic components of the space of diagonal coinvariants. As a consequence, we compute the action of tautological classes in HOMFLY-PT homology of $(n,n+1)$ torus knots and prove that it extends to an action of the Lie algebra of Hamiltonian vector fields on the plane. We also compute the action of differentials $d_N$ in Rasmussen spectral sequences from HOMLFY-PT to $\mathfrak{gl}(N)$ homology of $(n,n+1)$ torus knots.

Tautological classes for (n,n+1) torus knots

Abstract

We construct an explicit isomorphism between the HOMFLY-PT homology of torus knots and the direct sum of hook isotypic components of the space of diagonal coinvariants. As a consequence, we compute the action of tautological classes in HOMFLY-PT homology of torus knots and prove that it extends to an action of the Lie algebra of Hamiltonian vector fields on the plane. We also compute the action of differentials in Rasmussen spectral sequences from HOMLFY-PT to homology of torus knots.
Paper Structure (9 sections, 37 theorems, 91 equations, 2 figures)

This paper contains 9 sections, 37 theorems, 91 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Link homology
  4. Diagonal harmonics
  5. Hamiltonian vector fields
  6. Homology of torus links
  7. Full twist revisited
  8. From the full twist to $T(n,n+1)$
  9. Hooks and higher $a$-degrees

Key Result

Theorem 1.1

a) We have an isomorphism where $(\mathbf{x})=(x_1,\ldots,x_n)\subset \mathbb{C}[\mathbf{x}]$ is the maximal ideal. b) The cobordism map $\pi: \mathrm{HHH}^0(T(n,n))\to \mathrm{HHH}^0(T(n,n+1))$ is surjective and induces a grading-preserving isomorphism c) More generally, the cobordism map $\pi: \mathrm{HHH}(T(n,n))\to \mathrm{HHH}(T(n,n+1))$ is surjective and we have an isomorphism where $\mat

Figures (2)

  • Figure 1: Operators $\mathcal{F}_k$ and differentials for $T(3,4)$: $\mathcal{F}_1$ and $\mathcal{F}_2$ correspond to shorter and longer horizontal arrows; $d_1$ and $d_2$ correspond to solid and dotted diagonal arrows.
  • Figure 2: Action of $F_k^*$ on $\mathrm{DH}_3^{\mathrm{sgn}}$ (left) and action of $F_k$ on $\mathrm{DR}_3^{\mathrm{sgn}}$ (right)

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Conjecture 1.6
  • Remark 1.7
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • ...and 60 more