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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.

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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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  1. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Mustafa Hajij

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  1. Mustafa Hajij
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Correspondence to Mustafa Hajij.

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Hajij, M. The tail of a quantum spin network. Ramanujan J 40, 135–176 (2016). https://doi.org/10.1007/s11139-015-9705-9

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  • DOI: https://doi.org/10.1007/s11139-015-9705-9

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