Abstract
Nahm sums are q-series of a special hypergeometric type that appear in character formulas in the conformal field theory, and give rise to elements of the Bloch group, and have interesting modularity properties. In our paper, we show how Nahm sums arise naturally in the quantum knot theory - we prove the stability of the coefficients of the colored Jones polynomial of an alternating link and present a Nahm sum formula for the resulting power series, defined in terms of a reduced diagram of the alternating link. The Nahm sum formula comes with a computer implementation, illustrated in numerous examples of proven or conjectural identities among q-series. Primary 57N10; Secondary 57M25.
Highlights
Nahm sums are q-series of a special hypergeometric type that appear in character formulas in Conformal Field Theory, and give rise to elements of the Bloch group, and have interesting modularity properties
The colored Jones polynomial of a link is a sequence of Laurent polynomials in one variable with integer coefficients
We prove in full a conjecture concerning the stability of the colored Jones polynomial for all alternating links
Summary
The colored Jones polynomial of a link is a sequence of Laurent polynomials in one variable with integer coefficients. The limit of the colored Jones function of an alternating link leads us to consider generalized Nahm sums of the form (1). The theorem (proven in Section 11) shows the q-holonomicity of ΦK,n(q) for an alternating link, and gives a sharp improvement of the rate of convergence in the definition of stability. If K is a positive link, JK,n(q) is stable and the corresponding limit FK(x, q) is obtained by a Nahm sum associated to a positive downwards diagram of K. Section deduces the q-holonomicity of the sequence ΦK,k(q) of an alternating link from the q-holonomicity of the corresponding colored Jones polynomial. Zagier for their interest, encouragement and for the generous sharing of their ideas
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