Fray functors and equivalence of colored HOMFLYPT homologies

TL;DR

This paper develops functors on singular Soergel bimodules to relate different colored HOMFLYPT homologies, proving parity results for positive torus knots and advancing understanding of link invariants.

Contribution

It introduces new functors that connect various colored HOMFLYPT homologies, establishing their equivalence up to polynomial factors and resolving conjectures about knot homology parity.

Findings

Identifies equivalence of intrinsic and projector-colored homologies for links.

Establishes parity results for positive torus knots.

Partially resolves a conjecture on knot homology parity.

Abstract

We construct several families of functors on the homotopy category of singular Soergel bimodules that mimic cabling and insertion of column-colored projectors. We use these functors to identify the intrinsically-colored homology of Webster--Williamson and the projector-colored homology of Elias--Hogancamp for an arbitrary link, up to multiplication by a polynomial in the quantum degree $q$. Combined with the results of arXiv:2303.16271, this establishes parity results for the intrinsic column-colored homology of positive torus knots, partially resolving a conjecture of Hogancamp--Rose--Wedrich in arXiv:2107.09590.