Algorithms for Low-Dimensional Topology

TL;DR

This paper presents novel algorithms using AFL representations to efficiently compute key topological knot structures, including holonomic representations and the Kontsevich Integral, advancing computational methods in low-dimensional topology.

Contribution

It introduces new algorithms based on AFL representations for calculating knot invariants like holonomic forms and the Kontsevich Integral.

Findings

Efficient algorithm for holonomic knot representation

Method to compute the Kontsevich Integral using AFL

Potential for faster algorithms in low-dimensional topology

Abstract

In this article, we re-introduce the so called "Arkaden-Faden-Lage" (AFL for short) representation of knots in 3 dimensional space introduced by Kurt Reidemeister and show how it can be used to develop efficient algorithms to compute some important topological knot structures. In particular, we introduce an efficient algorithm to calculate holonomic representation of knots introduced by V. Vassiliev and give the main ideas how to use the AFL representations of knots to compute the Kontsevich Integral. The methods introduced here are to our knowledge novel and can open new perspectives in the development of fast algorithms in low dimensional topology.