A skein relation for singular Soergel bimodules

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 ${\mathfrak{sl}}_2$.

34 pages, many diagrams

References in corpus (2)

• 2-Kac-Moody algebras
• Homological perturbation theory with curvature