Knot invariants and higher representation theory II: the categorification of quantum knot invariants

TL;DR

This paper develops a framework for categorifying quantum knot invariants across all representations of quantum groups, extending previous specific cases and enabling the construction of knot homologies via categorified braiding and evaluation maps.

Contribution

It introduces a unified categorification approach for quantum knot invariants using Kac-Moody algebra representations and pictorial categorifications, generalizing prior results.

Findings

Categorified invariants match known invariants for sl(2), sl(3), and sl(n).

Categories are related by functors representing braiding and (co)evaluation maps.

Framework enables defining knot homologies from categorified quantum invariants.

Abstract

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by Mazorchuk-Stroppel and Sussan for sl(n). Our technique uses categorifications of the tensor product representations of Kac-Moody algebras and quantum groups, constructed a prequel to this paper. These categories are based on the pictorial approach of Khovanov and Lauda. In this paper, we show that these categories are related by functors corresponding to the braiding and (co)evaluation maps between representations of quantum groups. Exactly as these maps can be used to define quantum invariants attached to any tangle, their categorifications can be used to define knot homologies.