The quantum sl(n) graph invariant and a moduli space

TL;DR

This paper links a moduli space associated with colored trivalent graphs to the quantum sl(N) knot polynomial, showing that the Euler characteristic of this space equals the polynomial evaluated at one, thus connecting topology and algebraic geometry.

Contribution

It introduces a new moduli space framework for colored trivalent graphs and establishes its Euler characteristic as the quantum sl(N) polynomial at unity.

Findings

Euler characteristic of the moduli space equals the quantum sl(N) polynomial at one

Provides a geometric interpretation of the quantum sl(N) polynomial

Suggests potential extensions to the moduli space framework

Abstract

We associate a moduli problem to a colored trivalent graph; such graphs, when planar, appear in the state-sum description of the quantum sl(N) knot polynomial due to Murakami, Ohtsuki, and Yamada. We discuss how the resulting moduli space can be thought of a representation variety. We show that the Euler characteristic of the moduli space is equal to the quantum sl(N) polynomial of the graph evaluated at unity. Possible extensions of the result are also indicated.