Rectangular knot diagrams classification with deep learning

TL;DR

This paper explores using neural networks to classify rectangular knot diagrams and address the unknotting problem, leveraging finite combinatorial representations to understand neural network decision processes.

Contribution

It introduces a neural network approach to classify knot diagrams using rectangular Dynnikov diagrams, providing insights into neural network functioning on a mathematically well-understood problem.

Findings

Neural networks can distinguish knot types from rectangular diagrams.

The approach simplifies the unknotting problem to a finite combinatorial search.

Results demonstrate potential for neural networks in topological classification tasks.

Abstract

In this article we discuss applications of neural networks to recognising knots and, in particular, to the unknotting problem. One of motivations for this study is to understand how neural networks work on the example of a problem for which rigorous mathematical algorithms for its solution are known. We represent knots by rectangular Dynnikov diagrams and apply neural networks to distinguish a given diagram class from the given finite families of topological types. The data presented to the program is generated by applying Dynnikov moves to initial samples. The significance of using these diagrams and moves is that in this context the problem of determining whether a diagram is unknotted is a finite search of a bounded combinatorial space.