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

Kouki Yamaguchi

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.

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

Abstract

The -loop Kontsevich invariant of knots takes its value in the completion of the space of -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 -loop Kontsevich invariant of knots having the same Alexander polynomial. In this paper, we show that for the subspace generated by the image of the -loop Kontsevich invariant of genus knots having the same Alexander polynomial is finite dimensional. Further, we give some concrete calculations about those subspaces and dimensions in several simple cases.

Paper Structure

This paper contains 9 sections, 10 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Review of the Kontsevich invariant of knots
  3. The $n$-loop Kontsevich invariant of knots having the same Alexander polynomial
  4. A review of the rational version of the Aarhus integral and clasper surgeries
  5. A review of the rational version of the Aarhus integral
  6. A review of clasper surgeries
  7. Proofs of the theorems
  8. Proof of the fact that $\mathcal{V}(n,\Delta(t))$ is infinite dimensional
  9. Proof of the fact that $\mathcal{K}_{\leq g}^{\Delta(t)}$ is an infinite set unless it is $\emptyset$

Key Result

Theorem 3.1

For any integer $n\geq 2$, $g\geq 1$ and $\Delta(t)\in\mathcal{Z}$, the subspace $\mathcal{V}(n,g,\Delta(t))$ is finite dimensional. In particular, there exist finitely many elements $\beta_1,\cdots,\beta_d\in\widetilde{\mathcal{B}_{\text{conn}}^{(n)}}$ such that where $c_1,\cdots,c_d:\mathcal{K}_{\leq g}^{\Delta(t)}\to\mathbb{Q}$ are (restrictions of) Vassiliev invariants.

Theorems & Definitions (23)

  • Theorem 3.1
  • Corollary 3.2
  • Remark 3.3
  • Theorem 3.5
  • Corollary 3.6
  • Theorem 3.7
  • Remark 3.8
  • Theorem 3.9
  • Remark 3.10
  • Lemma 5.1
  • ...and 13 more