Skein modules, skein algebra and quantisation

Abstract

Skein modules are topological invariant for 3-manifolds constructed using knots and links. They are found to have important connections to important physical and mathematical problems. The main connection explored here is the quantisation of character variety, which has application in Physics as well as hyperbolic geometry. In this report, we will look at the work by Le, who developed a cutting map which allows one to describe certain classes of skein algebra as well as construct an important map called the quantum trace map. We will also look at two examples of skein module, two bridge knot complements and mapping torus, where we can describe the skeins using only part of the manifold.