On the Hilbert Polynomial of the HOMFLYPT Homology

TL;DR

This paper proves that the degree of the Hilbert polynomial of HOMFLYPT homology for a closed braid equals the number of its components minus one, revealing growth behavior and a hidden link polynomial.

Contribution

It establishes a precise relationship between the Hilbert polynomial degree and the number of components in HOMFLYPT homology, linking it to the HOMFLYPT polynomial.

Findings

Degree of Hilbert polynomial is l-1 for l-component braids

Hilbert polynomial controls homology growth with polynomial grading

Reveals a hidden link polynomial in HOMFLYPT polynomial

Abstract

We prove that the degree of the Hilbert polynomial of the HOMFLYPT homology of a closed braid $B$ is $l-1$, where $l$ is the number of components of $B$. This controls the growth of the HOMFLYPT homology with respect to its polynomial grading. The Hilbert polynomial also reveals a link polynomial hidden in the HOMFLYPT polynomial.