Rectangular knot diagrams classification with deep learning

Abstract

In this paper, we discuss applications of neural networks to recognizing knots and, in particular, to the unknotting problem. One of the 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’s class from a finite family 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. In this way, this paper provides a foundation for further work where the neural network itself will learn to use the Dynnikov moves for knot recognition. Source code of the programs is available at https://github.com/nerusskikh/deepknots .

Machine Learning Understands Knotted Polymers

“…Very recently the use of supervised ML approaches based on neural networks (NNs) has been recognized as a valuable tool in providing new insights into the mathematical problem of knot identification. − In ref , it was shown that a specific NN architecture, the long–short-term memory (LSTM) NN, can learn some global properties of the system that help with identifying the knotted state, once trained on sequences of bond directions along a semiflexible polymer ring of length N = 100.…”

Graph Neural Networks and 3-dimensional topology

“…We find that GEN+GAT model significantly outperforms the other models GCN+GAT and GCN+GCN. The model GCN+GAT seems to outperform GCN+GCN by few percent, but the performance difference is negligible 7 .…”

“…It is natural to apply GDL techniques also to mathematical problems in topology. In general, ML has been already used in various problems in low-dimensional topology, knot theory in particular, [3][4][5][6][7][8][9][10][11][12], as well as various physics-related problems in geometry (for a recent survey see [13]). However, the used neural network models were mostly not specific to GDL.…”