Faculty Mentor: Maggie Miller

Graduate Mentors: Remy Bohm and Aru Mukherjea

Project Description

Warm-up: diagrams of knotted surfaces

The first order of business in this project will be to learn about knotted surfaces in the \(4\)-sphere. You may know of classical knots, or embeddings of \(S^1\) into \(S^3\). We will work one dimension up, with embeddings of surfaces into \(S^4\), or knotted surfaces. One of the major problems with working with knotted surfaces is that there is no easy way to “draw” them, and so they have to be described using various types of diagrams.

The project therefore will begin with learning about a few different diagram systems for describing knotted surfaces. These include:

1. ch-diagrams, introduced by Yoshikawa¹. Ch-diagrams are similar to (classical) knot diagrams, but allow an additional type of crossing representing a saddle point in the surface.

2. banded unlink diagrams, developed by Kearton-Kurlin². These make use of a Morse function on the ambient space to describe the surface handle by handle, and look like an unlink with a bunch of bands (representing saddle points) attached and running through them.

3. triplane diagrams, which are due to Meier-Zupan³. (These two met when Jeffrey Meier was a grad student and Alex Zupan was a postdoc at UT!) Triplane diagrams look like three copies of the upper half plane with a set of arcs in each, which glue together to describe a surface in three pieces.

Surface-knot tabulation

When studying knots in any dimension, it is useful to have a table of what knots exist and what their properties are, for use as examples and test cases. Classical knots have been tabulated in a few different ways, for example see knotinfo.org. However, no such database exists for knotted surfaces. The overarching goal of this project would be to create a similar database to KnotInfo for knotted surfaces.

The current tabulation of knotted surfaces is due to Yoshikawa, listed in the same paper he introduces ch-diagrams. It consists of surfaces with ch-diagrams with up to \(10\) crossings of either kind. In total, this amounts to \(24\) surface-knots. An initial goal would be to use the same techniques to extend this tabulation, using whatever diagrammatic description from Part 1 is the most convenient.

We expect triplane diagrams to be the most easy to implement in a computer, as it shares some characteristics with braid diagrams of classical knots. Cahn-Matic-Ruppik⁴ have implemented various algorithms to compute invariants of triplane diagrams, which may be a useful starting point. [AAD⁺23]⁵ gives triplane diagrams for each knotted surface in the Yoshikawa table.

Suggested Background

At least one of the following:

• Point-set topology. More advanced topology or familiarity with knot theory or knotted surfaces a plus but not required.

• Coding experience (Python preferred, since we will likely be using SnapPy and SageMath for knot theory).

Application Form

Here is the link to the application form. Attached to the last question of the application form is a PDF containing the info above, in addition to a mini problem.

References