The lines of the Kontsevich integral and Rozansky's rationality conjecture

TL;DR

This paper develops techniques to analyze the loop expansion of the Kontsevich integral for knots, leading to progress on Rozansky's conjecture that organizes the integral into rationally labeled diagram series with bounded denominators.

Contribution

It introduces a new approach using the LMO invariant and surgery formulas to access the loop expansion, supporting Rozansky's conjecture on the rational structure of the Kontsevich integral.

Findings

Supports Rozansky's conjecture on rationality of the Kontsevich integral.

Provides a surgery formula applicable to the loop expansion analysis.

Develops new technology for analyzing knot invariants via the LMO invariant.

Abstract

This work develops some technology for accessing the loop expansion of the Kontsevich integral of a knot. The setting is an application of the LMO invariant to certain surgery presentations of knots by framed links in the solid torus. A consequence of this technology is a certain recent conjecture of Rozansky's. Rozansky conjectured that the Kontsevich integral could be organised into a series of ``lines'' which could be represented by finite $\Qset$-linear combinations of diagrams whose edges were labelled, in an appropriate sense, with rational functions. Furthermore, the conjecture requires that the denominator of the rational functions be at most the Alexander polynomial of the knot. This conjecture is obtained from an Aarhus-style surgery formula for this setting which we expect will have other applications.