On the n-loop Kontsevich invariant of knots having the same Alexander polynomial

TL;DR

This paper investigates the structure of the n-loop Kontsevich invariant for knots sharing the same Alexander polynomial, revealing finite dimensionality of certain subspaces and providing explicit calculations for specific cases.

Contribution

It demonstrates that for n≥2, the subspace generated by the n-loop invariant of knots with identical Alexander polynomial is finite dimensional and offers concrete calculations.

Findings

Subspace generated by the n-loop invariant is finite dimensional for n≥2.

Explicit calculations of these subspaces in simple cases.

The 1-loop part is determined by the Alexander polynomial.

Abstract

The $n$-loop Kontsevich invariant of knots takes its value in the completion of the space of $n$-loop open Jacobi diagrams, which is an infinite dimensional vector space. Since the 1-loop part is presented by the Alexander polynomial, we are interested in the image of the $\geq 2$-loop Kontsevich invariant of knots having the same Alexander polynomial. In this paper, we show that for $n\geq 2$ the subspace generated by the image of the $n$-loop Kontsevich invariant of genus $\leq g$ knots having the same Alexander polynomial is finite dimensional. Further, we give some concrete calculations about those subspaces and dimensions in several simple cases.