  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.
