On the 2–loop polynomial of knots

Abstract

The 2–loop polynomial of a knot is a polynomial characterizing the 2–loop part of the Kontsevich invariant of the knot. An aim of this paper is to give a methodology to calculate the 2–loop polynomial. We introduce Gaussian diagrams to calculate the rational version of the Aarhus integral explicitly, which constructs the 2–loop polynomial, and we develop methodology of calculating Gaussian diagrams showing many basic formulas of them. As a consequence, we obtain an explicit presentation of the 2–loop polynomial for knots of genus 1 in terms of derivatives of the Jones polynomial of the knots.

Corresponding to quantum and related invariants of 3–manifolds, we can formulate equivariant invariants of the infinite cyclic covers of knots complements. Among such equivariant invariants, we can regard the 2–loop polynomial of a knot as an “equivariant Casson invariant” of the infinite cyclic cover of the knot complement. As an aspect of an equivariant Casson invariant, we show that the 2–loop polynomial of a knot is presented by using finite type invariants of degree $\le 3$ of a spine of a Seifert surface of the knot. By calculating this presentation concretely, we show that the degree of the 2–loop polynomial of a knot is bounded by twice the genus of the knot. This estimate of genus is effective, in particular, for knots with trivial Alexander polynomial, such as the Kinoshita–Terasaka knot and the Conway knot.