Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rational Cherednik Algebras and Torus Knot Invariants

Xinchun Ma

Abstract

The HOMFLY polynomial of the $(m,n)$ torus knot $T_{m,n}$ can be extracted from the doubly graded character of the finite-dimensional representation $\mathrm{L}_{\frac{m}{n}}$ of the type $A_{n-1}$ rational Cherednik algebra as observed by Gorsky, Oblomkov, Rasmussen and Shende. It is furthermore conjectured that one can obtain the triply-graded Khovanov-Rozansky homology of $T_{m,n}$ by considering a certain filtration on $\mathrm{L}_{\frac{m}{n}}$. In this paper, we show that two of the proposed candidates, the algebraic filtration and the inductive filtration, are equal.

Rational Cherednik Algebras and Torus Knot Invariants

Abstract

The HOMFLY polynomial of the torus knot can be extracted from the doubly graded character of the finite-dimensional representation of the type rational Cherednik algebra as observed by Gorsky, Oblomkov, Rasmussen and Shende. It is furthermore conjectured that one can obtain the triply-graded Khovanov-Rozansky homology of by considering a certain filtration on . In this paper, we show that two of the proposed candidates, the algebraic filtration and the inductive filtration, are equal.
Paper Structure (24 sections, 46 theorems, 146 equations, 2 figures)

This paper contains 24 sections, 46 theorems, 146 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The rational Cherednik algebra and link homology
  3. The inductive filtration and the Hilbert scheme of points in $\mathbb{C}^2$
  4. Algebraic and geometric filtrations and the compactified Jacobian
  5. Representations of the rational Cherednik algebra
  6. Definitions
  7. c-Harmonic polynomials
  8. The power filtration and the inductive filtration
  9. The algebraic filtration and the power filtration
  10. The power filtration
  11. The Kazhdan filtration
  12. The inductive filtration
  13. Some first relations between $F^\mathfrak{a}$ and $F^\mathrm{ind}$
  14. Orthogonal lifts and the orthogonal complement of $I_{c-1}^W$
  15. Criteria for induction
  16. ...and 9 more sections

Key Result

Theorem 1.1

gors

Figures (2)

  • Figure 1: Filtrations from example \ref{['3,4']} where the numbers at the bottom indicate $\mathbf{h}$-weights
  • Figure 2: Filtrations from example \ref{['3,5']} where the numbers on the bottom indicate $\mathbf{h}$-weights

Theorems & Definitions (98)

  • Theorem 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Theorem 2.2
  • Example 2.3
  • Lemma 2.4
  • ...and 88 more