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.
Categories: math.RT math.CO