Mathematics – Geometric Topology
Scientific paper
2006-10-25
Communications in Contemporary Mathematics 10 (2008), suppl. 1, 1023-1032
Mathematics
Geometric Topology
9 pages, revised version of section 3 of math.GT/0604057, section 3.4 is new
Scientific paper
In this paper, by using the regulator map of Beilinson-Deligne on a curve, we show that the quantization condition posed by Gukov is true for the SL_2(C) character variety of the hyperbolic knot in S^3. Furthermore, we prove that the corresponding $\mathbb{C}^{*}$-valued closed 1-form is a secondary characteristic class (Chern-Simons) arising from the vanishing first Chern class of the flat line bundle over the smooth part of the character variety, where the flat line bundle is the pullback of the universal Heisenberg line bundle over $\mathbb{C}^{*}\times \mathbb{C}^{*}$. Based on this result, we give a reformulation of Gukov's generalized volume conjecture from a motivic perspective.
Li Weiping
Wang Qingxue
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