Braids, Complex Volume, and Cluster Algebra

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Abstract

We try to give a cluster algebraic interpretation of complex volume of knots. We construct the R-operator from the cluster mutations, and we show that it is regarded as a hyperbolic octahedron. The cluster variables are interpreted as edge parameters used by Zickert in computing complex volume.

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Vassiliev invariants and a strange identity related to the Dedekind eta-function

Don Zagier (2001)

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A Link Invariant from Quantum Dilogarithm

R. Kashaev (1995)

The link invariant, arising from the cyclic quantum dilogarithm via the particular \(R\)-matrix construction, is proved to coincide with the invariant of triangulated links in \(S^3\) introduced in R.M. Kashaev, Mod. Phys. Lett. A, Vol.9 No.40 (1994) 3757. The obtained invariant, like Alexander-Conway polynomial, vanishes on disjoint union of links. The \(R\)-matrix can be considered as the cyclic analog of the universal \(R\)-matrix associated with \(U_q(sl(2))\) algebra.

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Extended Bloch group and the Cheeger-Chern-Simons class

(2004)

We define an extended Bloch group and show it is naturally isomorphic to H_3(PSL(2,C)^\delta ;Z). Using the Rogers dilogarithm function this leads to an exact simplicial formula for the universal Cheeger-Chern-Simons class on this homology group. It also leads to an independent proof of the analytic relationship between volume and Chern-Simons invariant of hyperbolic 3-manifolds conjectured by Neumann and Zagier [Topology 1985] and proved by Yoshida [Invent. Math. 1985] as well as effective formulae for the Chern-Simons invariant of a hyperbolic 3-manifold.