Large color $R$-matrix for knot complements and strange identities

Abstract

The Gukov-Manolescu series, denoted by $F_K$, is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. In this paper we use the large color $R$-matrix to study $F_K$ for some simple links. Specifically, we give a definition of $F_K$ for positive braid knots, and compute $F_K$ for various knots and links. As a corollary, we present a class of `strange identities' for positive braid knots.

arXiv comment27 pages, 13 figures. v2 minor corrections

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• Higher rank $\hat{Z}$ and $F_K$
• Branches, quivers, and ideals for knot complements
• Cobordism invariants from BPS q-series
• Resurgent Analysis for Some 3-manifold Invariants
• BPS Invariants for a Knot in Seifert Manifolds
• Knot Complement, ADO-Invariants and their Deformations for Torus Knots
• The colored Jones polynomial and Kontsevich-Zagier series for double twist knots
• A Cable Knot and BPS-Series
• On knots, complements, and 6j-symbols
• A Cable Knot and BPS-Series II
• Inverted state sums, inverted Habiro series, and indefinite theta functions
• Non-Semisimple TQFT's and BPS $q$-Series
• A supergroup series for knot complements
• A Connect Sum Formula for the BPS Invariant of Knot Complements