Kashaev--Reshetikhin Invariants of Links

TL;DR

This paper clarifies the construction of Kashaev--Reshetikhin invariants for knots, showing their relation to character varieties and the A-polynomial, with computations at roots of unity.

Contribution

It provides detailed clarification of the Kashaev--Reshetikhin invariants and links them to geometric structures like the A-polynomial for hyperbolic knots.

Findings

Invariants can be viewed as functions on the character variety.

For hyperbolic knots, invariants relate to the A-polynomial curve.

Examples computed at a third root of unity.

Abstract

Kashaev and Reshetikhin previously described a way to define holonomy invariants of knots using quantum $\mathfrak{sl}_2$ at a root of unity. These are generalized quantum invariants depend both on a knot $K$ and a representation of the fundamental group of its complement into $\mathrm{SL}_2(\mathbb{C})$; equivalently, we can think of $\mathrm{KR}(K)$ as associating to each knot a function on (a slight generalization of) its character variety. In this paper we clarify some details of their construction. In particular, we show that for $K$ a hyperbolic knot $\mathrm{KaRe}(K)$ can be viewed as a function on the geometric component of the $A$-polynomial curve of $K$. We compute some examples at a third root of unity.