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International Mathematics Research Notices (2007) Vol. 2007 : article ID rnm022, 32 pages, doi:10.1093/imrn/rnm022 published on May 24, 2007
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Copyright © The Author 2007. Published by Oxford University Press.

Hecke Type Formula for Unified Witten–Reshetikhin–Turaev Invariants as Higher-Order Mock Theta Functions

Kazuhiro Hikami

Department of Physics, Graduate School of Science, University of Tokyo, Hongo 7–3–1, Bunkyo, Tokyo 113–0033, Japan

Correspondence: Correspondence to be sent to: Kazuhiro Hikami, Department of Physics, Graduate School of Science, University of Tokyo, Hongo 7–3–1, Bunkyo, Tokyo 113–0033, Japan. e-mail: hikami{at}phys.s.u-tokyo.ac.jp

We study the unified Witten–Reshetikhin–Turaev invariant for the Brieskorn homology sphere {Sigma}(2, 3, 6 p – 1) based on the cyclotomic expansion of the colored Jones polynomial for twist knot Kp. We discuss that the invariant has the same asymptotic expansion in N -> {infty} with the Ramanujan mock theta function when q is the root of unity q = exp(2 {pi}i/N), and that it can be regarded as the (6p – 1)-th order mock theta function. It is shown that it has the Hecke-type formula as in the case of the mock theta functions, though the quadratic form is positive definite while indefinite for almost all the Ramanujan mock theta functions.


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