Jakob Blaavand, Undergraduate Maths Seminar
In this talk we will introduce the basic notions in knot theory. We will start by defining what we mean by a knot and a link. A mathematical knot is almost the same as knots in the real world. If you tie a knot on a rope, you have to glue the ends of the rope back together again, so that your knot is on a circle. A link is just several knots, which might be linked together. We will also define, what it means that two knots are isomorphic.
Whenever you have defined some objects and isomorphisms between them, you want to classify them. That is, split the objects into disjoint classes, such that all objects of a class are isomorphic. In our case, given two knots can we specify whether they are isomorphic or not? We can define functions on the quotient space of isomorphic objects, and if the function gives different values on the two knots, they are definitely not isomorphic. A function defined on this moduli space, is called an invariant.
We will discuss one of the most famous invariants, the Jones polynomial. We will look at some basic properties of the Jones polynomial, and If time permits, we will discuss how to make the Jones polynomial into an invariant, which can separate the unknot from all other knots.
This talk will be understandable for everyone.