HOMFLYPT homology of Coxeter links

TL;DR

This paper explores the HOMFLYPT homology of Coxeter links, relating it to geometric structures on a generalized flag Hilbert scheme, thereby connecting knot invariants with algebraic geometry.

Contribution

It establishes a novel geometric interpretation of HOMFLYPT homology for Coxeter links using line bundles on generalized flag Hilbert schemes.

Findings

Identifies knot homology with sections of line bundles on flag Hilbert schemes.

Includes torus knots and links as special cases.

Provides a geometric framework for understanding knot homology.

Abstract

A Coxeter link is a closure of a product of two braids, one being a quasi-Coxeter element and the other being a product of partial full twists. This class of links includes torus knots \(T_{n,k}\) and torus links \(T_{n,nk}\). We identify the knot homology of a Coxeter link with the space of sections of a particular line bundle on a natural generalization of the punctual locus inside the flag Hilbert scheme of points in \(\mathbb{C}^2\).