THE MODULI PROBLEM OF LOBB AND ZENTNER AND THE COLORED 𝔰𝔩(N) GRAPH INVARIANT

Abstract

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