A neural network approach to predicting and computing knot invariants

TL;DR

This paper demonstrates that neural networks can accurately predict knot invariants such as quasipositivity, slice genus, and Ozsváth-Szabó τ-invariant, aiding in the classification and analysis of complex knots.

Contribution

It introduces a neural network-based method for predicting and computing key knot invariants, enabling the discovery of new quasipositive knots and improving computational efficiency.

Findings

Neural networks predict quasipositivity with high accuracy.

Identified 84 new quasipositive 11 and 12-crossing knots.

Predicted and computed slice genus and τ-invariant effectively.

Abstract

In this paper we use artificial neural networks to predict and help compute the values of certain knot invariants. In particular, we show that neural networks are able to predict when a knot is quasipositive with a high degree of accuracy. Given a knot with unknown quasipositivity we use these predictions to identify braid representatives that are likely to be quasipositive, which we then subject to further testing to verify. Using these techniques we identify 84 new quasipositive 11 and 12-crossing knots. Furthermore, we show that neural networks are also able to predict and help compute the slice genus and Ozsv\'{a}th-Szab\'{o} $\tau$-invariant of knots.