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A lift of the colored Jones polynomial of a knot

Stavros Garoufalidis, Campbell Wheeler

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 $Δ(t)$, to the recently introduced Habiro ring of the étale map $\mathbb{Z}[t^{\pm 1}]\to \mathbb{Z}[t^{\pm 1},Δ(t)^{-1}]$ (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.

A lift of the colored Jones polynomial of a knot

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 , to the recently introduced Habiro ring of the étale map (with Frobenius lifts for all primes ). 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.
Paper Structure (21 sections, 27 theorems, 140 equations, 1 figure)

This paper contains 21 sections, 27 theorems, 140 equations, 1 figure.

Table of Contents

  1. Introduction
  2. TQFT and the Habiro ring
  3. Rings for TQFT invariants
  4. Our results
  5. Extension to skein modules
  6. Knot polynomials associated to Lie algebras
  7. Plan of the proof
  8. Quantum knot invariants and rationality
  9. A state-sum for the colored Jones polynomial
  10. The $q\to 1$ limit, the Burau representation and the Alexander polynomial
  11. Rationality via the MacMahon's Master Theorem
  12. Further rationality via Lagrange inversion
  13. A review of the Habiro ring of an étale $\mathbb Z[t]$-algebra
  14. Basics on $p$-adic completions
  15. $\mathbb Z[t]$-étale algebras, their Habiro rings, and elements
  16. ...and 6 more sections

Key Result

Lemma 1.2

If the ring homomorphism $\mathrm{ev}_{t=1}:\mathbb Z[t^{\pm1}]\to\mathbb Z$ extends to a map $\mathrm{ev}_{t=1}:R\to\mathbb Z$, then there is an injective map Moreover, if $f(t,q) \in \mathcal{H}_{R/\mathbb Z[t^{\pm 1}]}$ satisfies $f(q^{n-1},q) \in \mathbb Z[q^{\pm 1}]$ for all $n>0$, then its image is in the subring given by $\varprojlim_n \mathbb Z[t^{\pm1},q^{\pm1}]/(t^{-1};q)_n$.

Figures (1)

  • Figure 1: Braid closure for $9_{44}$ with strand and crossing variables labeled. It also depicts the cycle associated with the crossing labeled by $k_4$.

Theorems & Definitions (59)

  • Definition 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Remark 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 49 more