Project IV 2023–24

What is an alternating knot?

Claudius Zibrowius

In knot theory, a knot diagram is called alternating if its crossings alternate over and under. For example, here are two diagrams:

The one on the left is alternating, the one on the right is not. A knot is called alternating if it admits an alternating knot diagram.

Of course, Ralph Fox knew this definition when he asked the above question. But he was unhappy with it. He was looking for an alternative, topological, diagram-free characterisation of this important class of knots.

In 2015, an answer to this long-standing question was found independently by two mathematicians, Joshua Howie [1] and Joshua Greene [2]. They characterised alternating knots in terms of spanning surfaces in the knot complement with certain homological properties.

The first goal of this project is to understand the statement and the proof of this characterisation. You will start by studying the following topics:

• links in three-manifolds and their linking numbers
• the Gordon-Litherland pairing
• link diagrams and chessboard surfaces

After completing the proof, you will be free to choose your preferred direction in which you continue your investigation. Here are some suggestions:

• Understand the detection algorithm for alternating knots given in the appendix of [2]. Here, you will learn about triangulations and normal surfaces, which play an important role in computational topology.
• Show that an alternating knot is fibred if and only if the leading coefficient of its Alexander polynomial is ±1. This is a well-known classical result in knot theory. The Alexander polynomial is one of the most famous knot invariants.
• What is an alternating tangle? Here, your task is to explore if the theorem of the two Joshuas can be extended to local versions of knots, so called tangles.

prerequisites

• Topology III

co-requisites

• Algebraic Topology IV

resources

• [1] Joshua Howie: A characterisation of alternating knot exteriors
• [2] Joshua Greene: Alternating links and definite surfaces
• [3] Lickorish: An introduction to Knot Theory