# Computer and Mathematical Sciences

## Mathematical Structures III A03

• Prof. Hitoshi Murakami
Keywordsknot theory, 3-manifold, low-dimensional topology, quantum topology

## Research on Knot Theory

My research interest is in knot theory.

A knot is a knotted circle embedded in the three-dimensional space as shown in the picture Fig. 1 below. You can see intuitively that the two knots are knotted, and that they are distinct. However, what would you do if you are asked to prove it mathematically?

In knot theory we associate polynomials (possibly with negative powers) to these knots. If these polynomials are different, then we conclude that these knots are different. The picture on the Fig. 2 shows how to compute the celebrated Jones polynomial. By using the recurrence formula, we can compute the Jones polynomials of the two knots Fig. 1, and see that these are indeed different. We often replace intuitive objects such as knots with algebraic objects such as polynomials in knot theory.

By removing a knot itself from the three-space (imagine that we remove a doughnut) and refilling it in another way (twist a doughnut and refill it), we can construct a totally new space. We call such a space a three-manifold. Note that this space is locally the same as the three-space that we live in.

• 7-7 knot (left), 8-16 knot (right) (Created by Mathematica)

• The recursion formula that defines the Jones polynomial V(K) of a knot K