Kashaev--Reshetikhin Invariants of Links

Abstract

Kashaev and Reshetikhin previously described a way to define holonomy invariants of knots using quantum 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 SL 2 (C); equivalently, we can think of 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 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. Contents 1. Introduction 1.1. Reshetikhin-Turaev and Kashaev-Reshetikhin invariants 1.2. Geometric interpretation 1.3. Relationship with other work Acknowledgements 2. Quantum sl 2 at a root of unity 2.1. The quantized function algebra 2.2. The Casimir and eigenvalues 3. Representations of the quantum group 4. Braiding on the quantized function algebra 5. Colored tangle diagrams 6. Construction of the invariant 7. The representation variety, the character variety, and the A-polynomial 8. Examples 8.1. The trefoil knot 8.2. The figure eight knot 8.3. Twist knots at the hyperbolic holonomy References