The Ramanujan Journal

, Volume 40, Issue 1, pp 135–176 | Cite as

The tail of a quantum spin network

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Abstract

The tail of a sequence \(\{P_n(q)\}_{n \in \mathbb {N}}\) of formal power series in \(\mathbb {Z}[[q]]\) is the formal power series whose first n coefficients agree up to a common sign with the first n coefficients of \(P_n\). This paper studies the tail of a sequence of admissible trivalent graphs with edges colored n or 2n. We use local skein relations to understand and compute the tail of these graphs. We also give product formulas for the tail of such trivalent graphs. Furthermore, we show that our skein theoretic techniques naturally lead to a proof for the Andrews–Gordon identities for the two-variable Ramanujan theta function as well to corresponding identities for the false theta function.

Keywords

Kauffman bracket skein modules Quantum invariants The colored Jones polynomial Trivalent graphs q-Series Identities of Rogers–Ramanujan type 

Mathematics Subject Classification

Primary 57M27 11P84 Secondary 57M15 

Notes

Acknowledgments

I would like to express my gratitude to Oliver Dasbach for his advice and patience. I am also grateful to Pat Gilmer for teaching me skein theory. I also appreciate the help of Moshe Cohen, Hany Hawasly, Kyle Isvan, Robert Osburn, Eyad Said, and Anastasiia Tsvietkova for offering a number of helpful comments.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA

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