Kashaev--Reshetikhin Invariants of Links

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.

Comments and questions welcome at [email protected]

References in corpus (1)

• Holonomy invariants of links and nonabelian Reidemeister torsion