Multiple Linking Number

TL;DR

This paper investigates the second Gauss diagram formula, called the multiple linking number, revealing its properties as a Vassiliev invariant and its sensitivity to Reidemeister moves, advancing link invariant theory.

Contribution

It introduces and analyzes the second Gauss diagram formula, the multiple linking number, highlighting its unique properties and sensitivity to Reidemeister moves.

Findings

One of the multiple linking numbers is a Vassiliev invariant.

The other multiple linking number is sensitive to the second Reidemeister move.

Gauss diagram formulas can detect subtle link properties.

Abstract

The linking number is the simplest link invariant given by Gauss; it is the first Gauss diagram formula expressed by one arrow among two circles. Proceeding the next stage, we study the second Gauss diagram formula consisting of two arrows among two circles. We call a function of this type the multiple linking number. There are two multiple linking numbers; one of them is ordinary Vassiliev invariant and the other function is surprisingly sensitive to the necessity of the second Reidemeister moves though any one-component Gauss diagram formula cannot detect the necessity.