SL₂(ℂ)-character variety of a hyperbolic link and regulator

Weiping Li and Qingxue Wang

Vol. 249 (2011), No. 2, 385–404

DOI: 10.2140/pjm.2011.249.385

Abstract

We analyze a special smooth projective variety Y ^h arising from some one-dimensional irreducible slices on the SL₂(ℂ)-character variety of a hyperbolic link in S³. We prove that a natural symbol obtained from these one-dimensional slices is a torsion in K₂(ℂ(Y ^h)). By using the regulator map from K₂ to the corresponding Deligne cohomology, we get some variation formulas on some Zariski open subset of Y ^h. From this we discuss a possible parametrized volume conjecture for both hyperbolic links and knots.

Keywords

Chern–Simons invariant, character variety, algebraic K-theory, hyperbolic links, volume conjecture, regulator map

Mathematical Subject Classification 2000

Primary: 57M25, 57M27

Secondary: 19F15, 14H50

Milestones

Received: 9 December 2009

Accepted: 9 September 2010

Published: 1 February 2011