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

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

2009-06-02

arxiv.org/abs/0906.0478v3

Pacific Journal of Mathematics 249 (2011), no. 2, 385-404

Mathematics

Geometric Topology

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

Scientific paper

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.

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