A skein relation for singular Soergel bimodules

TL;DR

This paper categorifies a skein relation for the HOMFLYPT invariant of links colored by one-column Young diagrams, linking it to singular Soergel bimodules and proving a related conjecture in quantum algebra.

Contribution

It introduces a categorification of the skein relation using homotopy equivalences of complexes built from singular Soergel bimodules, and proves a conjecture connecting the full twist complex with the categorical ribbon element.

Findings

Categorification of the colored skein relation achieved.

Homotopy equivalence between complexes constructed from Rickard complexes established.

Proved a conjecture relating the full twist complex to the categorical ribbon element.

Abstract

We study the skein relation that governs the HOMFLYPT invariant of links colored by one-column Young diagrams. Our main result is a categorification of this colored skein relation. This takes the form of a homotopy equivalence between two one-sided twisted complexes constructed from Rickard complexes of singular Soergel bimodules associated to braided webs. Along the way, we prove a conjecture of Beliakova--Habiro relating the colored 2-strand full twist complex with the categorical ribbon element for quantum $\mathfrak{sl}_2$.