On the geometry of two state models for the colored Jones polynomial

TL;DR

This paper establishes a natural bijection between two state models for the colored Jones polynomial, providing a new elementary proof of a known state-sum formula and extending it to links, revealing their fundamental equivalence.

Contribution

It introduces a bijection between states on different arc-graphs, unifying two models for the colored Jones polynomial and extending the formula to links.

Findings

The two state models are essentially equivalent.

The bijection simplifies understanding the state contributions.

The formula is extended to links with additional geometric insights.

Abstract

Using the flow property of the R-matrix defining the colored Jones polynomial, we establish a natural bijection between the set of states on the part arc-graph of a link projection and the set of states on a corresponding bichromatic digraph, called arc-graph, as defined by Garoufalidis and Loebl. We use this to give a new and essentially elementary proof for a knot state-sum formula of Garoufalidis and Loebl. We will show that the state-sum contributions of states on the part arc-graph defined by the universal R-matrix of $U_q(sl(2,\mathbb{C}))$ correspond, under our bijection of sets of states, to the contributions in the formula of Garoufalidis and Loebl. This will show that the two state models are in fact not essentially distinct. Our approach will also extend the formula of Garoufalidis and Loebl to links. This requires some additional non-trivial observations concerning the geometry of states on part arc-graphs. We will discuss in detail the computation of the arc-graph state-sum, in particular for 3-braid closures.