The Trivial Connection Contribution to Witten's Invariant and Finite Type Invariants of Rational Homology Spheres

TL;DR

This paper establishes a simple formula for the trivial connection contribution to Witten's invariant of rational homology spheres, linking it to finite type invariants and extending the analogy with Jones polynomial derivatives.

Contribution

It derives an analog of Melvin-Morton bounds and connects the trivial connection contribution to finite type invariants of rational homology spheres.

Findings

Provides a simple formula for trivial connection contribution to Witten's invariant.

Shows the n-th term in the expansion is a finite type invariant of specific order.

Establishes a manifold counterpart to the derivative-invariant relationship of Jones polynomial.

Abstract

We derive an analog of Melvin-Morton bound on the power series expansion of Jones polynomial of algebraically split links and boundary links. This allows us to produce a simple formula for the trivial connection contribution to Witten's invariant of rational homology spheres. We show that the n-th term in the 1/K expansion of the logarithm of this contribution is a finite type invariant of Ohtsuki order 3n and of at most Garoufalidis order n. This result is a manifold counterpart of the statement that n-th derivative of the Jones polynomial is Vassiliev's invariant of order n.