Rogers–Ramanujan type identities for alternating knots

Abstract

Abstract. We highlight the role of q-series techniques in proving identities arising from knot theory. In particular, we prove Rogers-Ramanujan type identities for alternating knots as conjectured by Garoufalidis, Lê and Zagier.

On certain identities involving Nahm-type sums with double poles

“…We also have identities for false theta functions due to Bringmann and one of us [6] (essentially the same identities were discovered in the analysis of 'tails' of colored Jones polynomials of (2, 2k) torus knots [12]; see also [5,16] for related identities).…”

q-series and tails of colored Jones polynomials

“…We first recall six q-series identities (see (2.1)-(2.3), Lemma 2.1, (4.3) and the proof of (4.1) in [10]). Namely,…”

“…where S K (q) is an explicitly constructed q-multisum (see pages 261-264 in [10]). Now, by Theorem 2 in [2], if T ′ K is the same as T ′ L for two alternating knots K and L, then Φ K (q) = Φ L (q).…”

“…In general, h b is a theta function if b is odd and a false theta function if b is even. Using q-series techniques, Keilthy and the second author [10] proved not only (1.1), but all of the remaining conjectural identities in [5].…”

“…The paper is organized as follows. In Section 2, we recall the necessary background from [10]. In Section 3, we prove Theorem 1.1.…”