The Moduli Problem of Lobb and Zentner and the Coloured sl(N) Graph Invariant

TL;DR

This paper extends the moduli problem introduced by Lobb and Zentner to coloured trivalent graphs, linking representation varieties to quantum invariants and generalizing previous results for two-colour cases.

Contribution

It generalizes the moduli space construction to arbitrary colourings by irreducible anti-symmetric representations of sl(N), broadening the connection between topology and algebraic geometry.

Findings

Representation variety interpretation for coloured graphs

Euler characteristic matches the sl(N) polynomial at 1

Extension from two-colour to arbitrary colourings

Abstract

Motivated by a possible connection between the $\mathrm{SU}(N)$ instanton knot Floer homology of Kronheimer and Mrowka and $\mathfrak{sl}(N)$ Khovanov-Rozansky homology, Lobb and Zentner recently introduced a moduli problem associated to colourings of trivalent graphs of the kind considered by Murakami, Ohtsuki and Yamada in their state-sum interpretation of the quantum $\mathfrak{sl}(N)$ knot polynomial. For graphs with two colours, they showed this moduli space can be thought of as a representation variety, and that its Euler characteristic is equal to the $\mathfrak{sl}(N)$ polynomial of the graph evaluated at 1. We extend their results to graphs with arbitrary colourings by irreducible anti-symmetric representations of $\mathfrak{sl}(N)$.