Patrick Kinnear

Project Title: Skein Module Topological Field Theory 

Geometry and topology are the mathematical study of shapes. While mathematicians study shapes as purely abstract objects, we often use our intuition from living in the physical world to guide us. The area of topological field theory uses the formal mathematical tools developed to explain quantum physics to understand the study of shape.

An invariant is a way of associating a mathematical object to each shape so that if two shapes have different objects associated to them, then the shapes must be different. Topological field theory leads to an invariant called a skein module. This object is obtained by considering all the ways we could place a collection of knots inside our shape.

This object seems like it could be very large. The physicist Edward Witten, based on physical considerations, conjectured that for 3d shapes, the skein module would not be too big: that the number mathematicians use to describe its size, the dimension, would be a finite number. Recently Gunningham, Jordan and Safronov gave a mathematical proof of Witten’s conjecture, showing that skein modules for 3d shapes usually have finite dimension.

In this project, we will explore skein modules further. First, we will compute the dimension of the skein module for some interesting shapes. Second, it is known that there are some conditions in which skein modules are not finite dimensional: we will attempt to describe precisely when this is the case. Finally, we will explore the relationship of skein modules with other celebrated invariants, particularly those with connections to physics.