Communications in Number Theory and Physics

Volume 9 (2015)

Number 3

Evaluation of state integrals at rational points

Pages: 549 – 582



Stavros Garoufalidis (School of Mathematics, Georgia Institute of Technology, Atlanta, Ga. U.S.A.)

Rinat Kashaev (Section de Mathématiques, Université de Genève, Switzerland)


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.


state-integrals, $q$-series, quantum dilogarithm, cyclic dilogarithm, Rogers dilogarithm, quasi-periodic functions, Nahm equation, gluing equations, $4_1$, $5_2$ and $(-2, 3, 7)$ pretzel knot

Published 11 September 2015