Evaluation of state integrals at rational points

Stavros Garoufalidis; Rinat Kashaev

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arXiv · 1411.6062

Evaluation of state integrals at rational points

Stavros Garoufalidis·Rinat Kashaev

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Abstract

Multi-dimensional state-integrals of products of Faddeev's quantum dilogarithms arise frequently in Quantum Topology, quantum Teichmüller theory and complex Chern--Simons theory. Using the quasi-periodicity property of the quantum dilogarithm, we evaluate 1-dimensional state-integrals at rational points and express the answer in terms of the Rogers dilogarithm, the cyclic (quantum) dilogarithm and finite state-sums at roots of unity. We illustrate our results with the evaluation of the state-integrals of the $4_1$, $5_2$ and $(-2,3,7)$ pretzel knots at rational points.

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Stavros Garoufalidis, Rinat Kashaev. 2015-04-27. Evaluation of state integrals at rational points. https://arxiv.org/abs/1411.6062

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