A combinatorial approach to functorial quantum sl(k) knot invariants

TL;DR

This paper presents a combinatorial and categorified approach to sl(k) knot invariants using category O and foams, aiming for elementary proofs and a representation-theoretic interpretation of sl(k)-homology.

Contribution

It introduces a new combinatorial functorial invariant for webs and links, extending to sl(k)-homology via derived categories, with potential links to Khovanov-Rozansky homology.

Findings

Constructed an exact functor invariant for webs satisfying MOY relations

Extended the invariant to sl(k)-links using derived categories

Proposed a representation-theoretic interpretation of sl(k)-homology

Abstract

This paper contains a categorification of the sl(k) link invariant using parabolic singular blocks of category O. Our approach is intended to be as elementary as possible, providing combinatorial proofs of the main results of Sussan. We first construct an exact functor valued invariant of webs or 'special' trivalent graphs labelled with 1, 2, k-1, k satisfying the MOY relations. Afterwards we extend it to the sl(k)-invariant of links by passing to the derived categories. The approach using foams appears naturally in this context. More generally, we expect that our approach provides a representation theoretic interpretation of the sl(k)-homology, based on foams and the Kapustin-Lie formula. Conjecturally this implies that the Khovanov-Rozansky link homology is obtained from our invariant by restriction.