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A refined 1-cocycle for regular isotopies and the refined tangle equations

Thomas Fiedler

Abstract

We refine the combinatorial 1-cocycle $\mathbb{L}R_{reg}$ for regular isotopies of long knots to a 1-cocycle with values in the free $\mathbb{Z}[x,x^{-1}]$-module generated by regular isotopy classes of oriented tangles with exactly one signed ordinary double point. We use it to define the refined tangle equations for couples of knot diagrams, where the coefficients are now Laurent polynomials instead of integers. A solution of the tangle equations gives quantitative information about any knot isotopy which relates two given knot diagrams. If the tangle equations have no solution, then the diagrams represent different knots.

A refined 1-cocycle for regular isotopies and the refined tangle equations

Abstract

We refine the combinatorial 1-cocycle for regular isotopies of long knots to a 1-cocycle with values in the free -module generated by regular isotopy classes of oriented tangles with exactly one signed ordinary double point. We use it to define the refined tangle equations for couples of knot diagrams, where the coefficients are now Laurent polynomials instead of integers. A solution of the tangle equations gives quantitative information about any knot isotopy which relates two given knot diagrams. If the tangle equations have no solution, then the diagrams represent different knots.
Paper Structure (5 sections, 2 theorems, 22 figures)

This paper contains 5 sections, 2 theorems, 22 figures.

Table of Contents

  1. Introduction and main results
  2. The refined 1-cocycles
  3. Proof
  4. The refined tangle equations
  5. Adding the longitude breaks the telescoping effect

Key Result

Proposition 1

There is a combinatorial 1-cocycle $\mathbb{L}R_{reg}(x) \in Z^1(M; H_0(M_+;\mathbb{Z}[x,x^{-1}]) \bigoplus H_0(M_-;\mathbb{Z}[x,x^{-1}]))$. Moreover, if $D$ and $D'$ are regularly isotopic, then $\mathbb{L}R_{reg}(x)(push(K,D)) - \mathbb{L}R_{reg}(x)(push(K,D'))=$$K \sum_i D_i(\sum_j a_{i,j}(x^{b

Figures (22)

  • Figure 1: The names of the crossings in a R III-move
  • Figure 2: The six global types of triple crossings e.g. $r_a$ means the type on the right with $a=\infty$ and $l_c$ means the type on the left with $c=\infty$.
  • Figure 3: Local types of a triple crossing together with the local side of the discriminant. The triple crossings of the types 2 and 6 are called star-like.
  • Figure 4: the f-crossings for $d$
  • Figure 5: crossings of type 1 which are not f-crossings for $d$
  • ...and 17 more figures

Theorems & Definitions (8)

  • Proposition 1
  • Theorem 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 1
  • Remark 2