Question is related to knot theory

Knot theory

Did you tie your shoe laces the same way today as yesterday? We are all familiar with knots from everyday life. But when does a mathematician consider two knots to be the same? To capture the mathematical, rather than physical, features of a knot, we glue the ends together to form a circular loop, and then treat the string as infinitely strong, thin, and stretchy. The downside of doing so is that it is not immediately clear how to distinguish two knots, as there are infinitely many configurations for the same knot.

For this project you should find out about how knots are represented, and about some of the invariants used to tell them apart. Some possible areas of investigation include:

• tricolourability.

• genus.

• the Jones polynomial.

You should include plenty of your own examples to illustrate your under- standing.

Prerequisites

No particular background knowledge is needed, but you will need to be able to interpret a 2D diagram as a 3D object in space and visualise it moving over time.

Literature

• Colin Adams, The Knot Book

• Peter Cromwell, Knots and Links