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).

A generators and relations description of a representation category of Uq(𝔤𝔩(1|1))

“…In fact, it is enough to specialise to colours in f1; 2g, by the same argument as [9,Proposition 5.2]. However, of course, since all closed diagrams reduce to circles after applying a sequence of MOY moves, this link invariant will be identically 0.…”

Spider evaluation and representations of web groups

“…Since T is a rooted tree, there is a unique maximal geodesic for each state. To compute the contribution of a state to the spider evaluation of Γ start with [3] c where c is the number of components of the state, and multiply it times [2] b where b is the number of bubbles that were collapsed along the maximal geodesic corresponding to the state. The sum over all states of these contributions is the spider evaluation of the web.…”

“…The circle can be oriented counterclockwise or clockwise. The symmetrized Poincaré polynomial of CP (2) is [3].…”

“…A more general study of representation spaces of colored graphs was undertaken in [11,3]. The work of Lobb and Zentner is for sl(N ).…”

Junctions of surface operators and categorification of quantum groups

“…• The analysis of section 3 suggests to carry out the analogue of [63,64] for the complexified gauge group G C and to compare Poincaré polynomials of the resulting moduli spaces M(G C , Γ) to MOY polynomials of the planar graphs Γ.…”