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A Contribution of the Trivial Connection to Jones Polynomial and Witten's Invariant of 3d Manifolds I

Lev Rozansky

TL;DR

Using a path integral formulation of 3d Chern-Simons theory, the work derives a precise relation between the Jones polynomial and the Alexander polynomial via Reidemeister-Ray-Singer torsion, and proves a conjectured $1/K$-expansion limitation. A key result is a surgery formula for the trivial-connection contribution to Witten's invariant on rational homology spheres, with the $2$-loop term coinciding with Walker's Casson-Walker invariant. The study further extends the analysis to Seifert manifolds and discusses non RHS cases, revealing a split of the trivial-connection contribution into a finite polynomial part and an asymptotic $K^{-1}$ series. The findings illuminate how low-order loop corrections encode classical 3-manifold invariants and hint at structured connections to Vassiliev invariants through knot complements.

Abstract

We use the Chern-Simons quantum field theory in order to prove a recently conjectured limitation on the 1/K expansion of the Jones polynomial of a knot and its relation to the Alexander polynomial. This limitation allows us to derive a surgery formula for the loop corrections to the contribution of the trivial connection to Witten's invariant. The 2-loop part of this formula coincides with Walker's surgery formula for Casson-Walker invariant. This proves a conjecture that Casson-Walker invariant is a 2-loop correction to the trivial connection contribution to Witten's invariant of a rational homology sphere. A contribution of the trivial connection to Witten's invariant of a manifold with nontrivial rational homology is calculated for the case of Seifert manifolds.

A Contribution of the Trivial Connection to Jones Polynomial and Witten's Invariant of 3d Manifolds I

TL;DR

Using a path integral formulation of 3d Chern-Simons theory, the work derives a precise relation between the Jones polynomial and the Alexander polynomial via Reidemeister-Ray-Singer torsion, and proves a conjectured -expansion limitation. A key result is a surgery formula for the trivial-connection contribution to Witten's invariant on rational homology spheres, with the -loop term coinciding with Walker's Casson-Walker invariant. The study further extends the analysis to Seifert manifolds and discusses non RHS cases, revealing a split of the trivial-connection contribution into a finite polynomial part and an asymptotic series. The findings illuminate how low-order loop corrections encode classical 3-manifold invariants and hint at structured connections to Vassiliev invariants through knot complements.

Abstract

We use the Chern-Simons quantum field theory in order to prove a recently conjectured limitation on the 1/K expansion of the Jones polynomial of a knot and its relation to the Alexander polynomial. This limitation allows us to derive a surgery formula for the loop corrections to the contribution of the trivial connection to Witten's invariant. The 2-loop part of this formula coincides with Walker's surgery formula for Casson-Walker invariant. This proves a conjecture that Casson-Walker invariant is a 2-loop correction to the trivial connection contribution to Witten's invariant of a rational homology sphere. A contribution of the trivial connection to Witten's invariant of a manifold with nontrivial rational homology is calculated for the case of Seifert manifolds.

Paper Structure

This paper contains 5 sections, 10 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Jones Polynomial and Reidemeister-Ray-Singer Torsion
  3. A Trivial Connection Contribution to Witten's Invariant
  4. Beyond the Rational Homology Spheres
  5. Discussion

Key Result

Proposition 2.1

If the knot ${\cal K}$ is canonically framed (i.e. the linking number $\nu$ between the cycle $C_2$ which determines the framing and ${\cal K}$ is zero), then Moreover, here $\Delta_A(S^3,{\cal K};\exp(2\pi ia))$ is the Alexander polynomial of ${\cal K}$ normalized in such a way that $\Delta(S^3,{\rm unknot};\exp(2\pi i a))=1$, $\Delta_A(M,{\cal K};\exp(2\pi ia))$ is real.

Theorems & Definitions (10)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 3.1
  • Corollary 3.1
  • Proposition 3.2
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition A.1