Spin Link Homology

TL;DR

This paper introduces a new involution on certain Khovanov--Rozansky homologies that yields link invariants categorifying spin-colored quantum polynomials for specific Lie algebras, advancing link homology theory.

Contribution

It develops a novel involution on $ ext{sl}_{2n}$ Khovanov--Rozansky homology that produces link invariants categorifying spin-colored $ ext{so}_{2n+1}$ polynomials, connecting with quantum webs and $ ext{iota}$quantum groups.

Findings

Involutions produce link invariants for $n=1,2,3$.

Categorification of spin-colored $ ext{so}_{2n+1}$ polynomials.

Partial development of quantum $ ext{so}_{2n+1}$ web theory.

Abstract

We put a new spin on Khovanov--Rozansky homology. That is, we equip $\Lambda^n$-colored $\mathfrak{sl}_{2n}$ Khovanov--Rozansky homology with an involution whose $\pm 1$-eigenspaces are link invariants. When $n=1,2,3$ (and assuming technical conjectures for $n \geq 4$), we prove that this refined invariant categorifies the spin-colored $\mathfrak{so}_{2n+1}$ quantum link polynomial. Along the way, we partially develop the theory of quantum $\mathfrak{so}_{2n+1}$ webs and make contact with $\iota$quantum groups.