$S{L}_2(\mathbb{C})$-Character Variety of a Hyperbolic Link and Regulator

Abstract

In this paper, we study the $S{L}_2(\mathbb{C})$ character variety of a hyperbolic link in $S^3$. We analyze a special smooth projective variety $Y^h$ arising from some 1-dimensional irreducible slices on the character variety. We prove that a natural symbol obtained from these 1-dimensional slices is a torsion in $K_2(\mathbb{C}(Y^h))$. By using the regulator map from $K_2$ to the corresponding Deligne cohomology, we get some variation formulae on some Zariski open subset of $Y^h$. From this we give some discussions on a possible parametrized volume conjecture for both hyperbolic links and knots.

v.2 combined the final version with v.1 by mistake. v.3 is the final version, 17 pages