Rational Cherednik Algebras and Torus Knot Invariants

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.