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arXiv · 2402.18770

Rational Cherednik Algebras and Torus Knot Invariants

Xinchun Ma
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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.

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Xinchun Ma. 2024-02-29. Rational Cherednik Algebras and Torus Knot Invariants. https://arxiv.org/abs/2402.18770

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