Slope and concordance of links

TL;DR

This paper studies the slope invariant of colored links, proving its invariance under concordance and providing a new computational formula using C-complexes and Seifert forms.

Contribution

It establishes the invariance of the slope under colored concordance and introduces a formula for its calculation via C-complexes and Seifert forms.

Findings

Slope is invariant under colored concordance.

A new formula for computing the slope using C-complexes.

Connections between the slope and classical link invariants.

Abstract

The slope is an isotopy invariant of colored links with a distinguished component, initially introduced by the authors to describe an extra correction term in the computation of the signature of the splice. It appeared to be closely related to several classical invariants, such as the Conway potential function or the Kojima eta-function (defined for two-components links). In this paper, we prove that the slope is invariant under colored concordance of links. Besides, we present a formula to compute the slope in terms of C-complexes and generalized Seifert forms.