Knot Complement, ADO Invariants and their Deformations for Torus Knots

Abstract

A relation between the two-variable series knot invariant and the Akutsu-Deguchi-Ohtsuki (ADO) invariant was conjectured recently. We reinforce the conjecture by presenting explicit formulas and/or an algorithm for particular ADO invariants of torus knots obtained from the series invariant of complement of a knot. Furthermore, one parameter deformation of ADO3 polynomial of torus knots is provided.

Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants

“…The simplest non-trivial example of a non-positive braid knot is the figure-8 knot K = 4 1 that will be one of our examples below. Further evidence for the Conjecture 3 was found in [80] soon after a preprint of this paper appeared. The Conjecture 3 is also motivated from the physics perspective as follows.…”

Non-Semisimple TQFT's and BPS q-Series

“…Then, using identities of the form (2.28) we arrive at the desired relation between (2.27) and (2.1). Based on this relation between the R-matrices, it is not surprising to expect the following conjectural relation between the corresponding knot invariants [17,39]…”

“…This is especially surprising given a large number of close ties that WRT invariants of knots and 3manifolds have with quantum field theory and string theory. The situation started to change about a year ago [39] and we hope that the present paper can be another step toward bridging this gap, see also [16,17,19,21,26,27,34,35,68] for closely related work.…”

“…The space of such local operators is the space of states on Σ = S 2 \{p 1 , p 2 }; in particular, it is non-trivial for BCGP theory indicating the non-semisimple nature of the TQFT. As a special case, the space of local operators at the junction of two trivial lines is simply H S 2 , and for GPPV theory its structure is conveniently encoded in the unreduced 17 version of the Z-invariant for S 1 × S 2 : 1 2…”

Witt invariants from q-series $\hat{Z}$

“…Later, it was shown in [7] that this series invariant F K in turn are connected to the Akutsu-Deguchi-Ohtsuki (ADO) polynomials [1]. This conjecture was reinforced in [2]. Recently, another facet of Ẑ was revealed in [9].…”