The simplicial-volume form of Kashaev's conjecture

For any link LL, let JN(L)J_N(L) be the colored Jones polynomial evaluated at exp(2π1/N)\exp\left(2\pi\sqrt{-1}/N\right). Let L\Vert L\Vert denote the simplicial volume of the complement of LL, and let v3v_3 be the volume of the ideal regular tetrahedron. The simplicial-volume conjecture. For every link LL,

L=2πv3limNlogJN(L)N.\Vert L\Vert=\frac{2\pi}{v_3}\lim_{N\to\infty}\frac{\log|J_N(L)|}{N}.

This is presented as an extension of Kashaev's volume conjecture. The formulation is not valid for links in general because JN(L)J_N(L) vanishes for a split link; for hyperbolic links, simplicial volume is related to hyperbolic volume by multiplication by v3v_3.

Sources & referencesView supporting material

Primary source

Hitoshi Murakami, Jun Murakami, Miyuki Okamoto, Toshie Takata and Yoshiyuki Yokota, “Kashaev's conjecture and the Chern-Simons invariants of knots and links”, arXiv:math/0203119 (2002).

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