Title: Neural Knots: A Machine Learning Approach to Knot Theory
Abstract: In recent years artificial neural networks have received a great deal of attention due to their ability to detect subtle and very complex patterns in large data sets. They have become an important machine learning tool and have been applied successfully to many problems, including computer vision, fraud detection, medical diagnostics, and financial modeling.
Knots are topological objects which were first studied in the 1700s. They appear in modern physics and biology, and play a crucial role in the study of the topology of 3 and 4-dimensional manifolds. Numerous algebraic, geometric, and analytic invariants can be defined which reflect interesting properties of these knots.
In this talk I will outline an approach to using neural networks to model knot invariants. Although these networks can be used to model a variety of invariants, we will focus on two which are particularly difficult to compute: quasipositivity and the slice genus. After defining these invariants and discussing their importance in low-dimensional topology we will show how they can be computed and studied using predictions made from neural networks. We will also outline an approach to using these techniques to uncover new algebraic structure in the related knot concordance group.