A lift of the colored Jones polynomial of a knot

TL;DR

This paper extends the colored Jones polynomial of a knot to the Habiro ring, enabling new insights into quantum invariants, loop expansions at roots of unity, and connections to Lie algebra representations.

Contribution

It introduces a novel lift of the colored Jones polynomial to the Habiro ring, confirming a conjecture and enabling new expansions and generalizations.

Findings

Existence of a loop expansion at roots of unity.

Lift of Ohtsuki invariants to the Habiro ring.

Natural extensions to skein modules and Lie algebra colorings.

Abstract

Habiro lifted the Witten-Reshetikhin-Turaev invariant of an integer homology 3-sphere (a complex-valued function on the set of complex roots of unity) to an element of the Habiro ring. We lift the colored Jones polynomial of a knot, with Alexander polynomial $\Delta(t)$, to the recently introduced Habiro ring of the \'etale map $\mathbb{Z}[t^{\pm 1}]\to \mathbb{Z}[t^{\pm 1},\Delta(t)^{-1}]$ (with Frobenius lifts $t\mapsto t^p$ for all primes $p$). This implies the existence of a loop expansion at roots of unity (confirming a conjecture of Habiro), and a lift of power series invariants of Ohtsuki for 3-manifolds with Betti number 1 to a Habiro ring. Our results have natural extensions to the skein module of a knot complement, and they suggest a natural lift of the colored Jones polynomial colored by representations of a simple Lie (super) algebra.