Variations of Milnor's triple linking number

TL;DR

This paper introduces simple Gauss diagram formulas for Milnor-type Vassiliev invariants, providing a new approach to compute topological invariants of polymers more efficiently.

Contribution

It presents novel Gauss diagram formulas for Milnor's triple linking number, extending the computational tools for topological invariants.

Findings

Gauss diagram formulas for Milnor's invariants are established

Formulas are non-torsion valued, unlike traditional Milnor invariants

Provides a new method for analyzing topological polymers

Abstract

Topological polymers have various topological types, and they are expressed by graphs. However, the Jones polynomial, we have a difficulty to compute it; computational time is growing exponentially with respect to the crossing number. The simplest Vassiliev invariant is the linking number and thus we will seek a next simple one is as the Milnor's triple linking number. In this paper, we introduce simple Gauss diagram formulas of Vassiliev invariants of Milnor type. These are non-torsion valued, whereas the base-point-free Milnor's triple linking number is usually torsion-valued.