$SL_2(\mathbb{C})$-Character Variety of a Hyperbolic Link and Regulator

TL;DR

This paper investigates the $SL_2( ext{C})$ character variety of hyperbolic links, revealing torsion properties in $K_2$, deriving regulator variation formulas, and discussing implications for a parametrized volume conjecture.

Contribution

It introduces a novel analysis of the character variety via $K_2$ torsion and regulator maps, offering new insights into hyperbolic link invariants and volume conjectures.

Findings

The symbol from 1D slices is torsion in $K_2$.

Derived regulator variation formulas on $Y^h$.

Discussed potential parametrized volume conjecture.

Abstract

In this paper, we study the $SL_2(\mathbb{C})$ 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.