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Slopes of links and signature formulas

Alex Degtyarev, Vincent Florens, Ana G. Lecuona

Abstract

We present a new invariant, called slope, of a colored link in an integral homology sphere and use this invariant to complete the signature formula for the splice of two links. We develop a number of ways of computing the slope and a few vanishing results. Besides, we discuss the concordance invariance of the slope and establish its close relation to the Conway polynomials, on the one hand, and to the Kojima--Yamasaki $η$-function (in the univariate case) and Cochran invariants, on the other hand.

Slopes of links and signature formulas

Abstract

We present a new invariant, called slope, of a colored link in an integral homology sphere and use this invariant to complete the signature formula for the splice of two links. We develop a number of ways of computing the slope and a few vanishing results. Besides, we discuss the concordance invariance of the slope and establish its close relation to the Conway polynomials, on the one hand, and to the Kojima--Yamasaki -function (in the univariate case) and Cochran invariants, on the other hand.

Paper Structure

This paper contains 13 sections, 7 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The signature formula
  3. Signature of colored links
  4. Splice of colored links
  5. (Non)-additivity of the signature
  6. The slope
  7. Definition of the slope
  8. Fox calculus
  9. The rationality
  10. Slopes via $C$-complexes
  11. Concordance invariance
  12. Slopes and concordance
  13. Two-component links in the sphere

Key Result

proposition 1

Let $\Go \in \mathcal{U}(K/L):=\mathcal{A}(K/L)\cap(S^1 \sminus 1)^\mu$ be a unitary character. Then $\dim Z(\Go)=1$ and, hence, $(K/L)(\Go)$ is well defined. Moreover, $(K/L)(\Go) \in \mathbb{R} \cup \infty$.

Figures (2)

  • Figure : The correction terms $\Delta\sigma$ (left) and $\Delta n$ (right) in \ref{['th.main']}.
  • Figure : The links $K \cup L_n\subset S^3$ in \ref{['ex.equal.mu']}.

Theorems & Definitions (11)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • proposition 1: see DFL2
  • proposition 2: essentially, DFL2
  • corollary 1: see DFL2
  • corollary 2
  • corollary 3
  • corollary 4
  • ...and 1 more