A skein relation for singular Soergel bimodules

2021/07/16 by Matthew Hogancamp, Hogancamp, Matthew, David E. V. Rose +3 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2107.08117

openalex publication_date 2021/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \mathfraksl₂.

Cited by

• Doubly-graded exponential growth of colored torus knot homology
• The minimal Rickard complexes of braids on two strands

Related

• Fray functors and equivalence of colored HOMFLYPT homologies
• Congruent skein relations for colored HOMFLY-PT invariants and colored Jones polynomials
• Holonomy braidings, biquandles and quantum invariants of links with SL₂(\mathbb C) flat connections
• Link splitting deformation of colored Khovanov--Rozansky homology
• The minimal Rickard complexes of braids on two strands