Project Description

Faculty Mentor: Maggie Miller

Graduate Mentors: Remy Bohm and Aru Mukherjea

Drawing the Gordian Graph

A knot is an embedding of \(S^1\) into \(S^3\). Knots are represented by knot diagrams, which are projections of the knot onto a plane with intersections labeled with “over” and “under” strands. Every knot has infinitely many diagrams, and determining how knots relate to their diagrams is a major guiding question in knot theory.

We say that two knots \(K_1\) and \(K_2\) are related by crossing change if there exists a diagram \(D_1\) of \(K_1\) and a crossing in that diagram such that if we change which strand is the “over” strand in that diagram, we get a diagram \(D_2\) for \(K_2\). This is related to the unknotting number of a knot, which is the minimal number of crossing changes necessary to unknot a knot, minimized over all possible diagrams. In general, this is hard to compute, and there exist \(10\)-crossing knots for which the unknotting number is still unknown.

However, we can also ask which knots are related to each other by crossing change. For instance, nontrivial knots \(K_1\) and \(K_2\) might both have diagrams that are only one crossing change away from a diagram for the unknot (and hence have unknotting number one), but that does not imply that \(K_1\) and \(K_2\) are related to each other by a single crossing change.

The data of how knots are related by crossing change is usually described as the Gordian graph. The Gordian graph is constructed as follows. The vertices are (isotopy classes of) knots. Two vertices \(K_1\) and \(K_2\) are joined by an edge if there is some diagram of \(K_1\) where a single crossing change turns it into a diagram for \(K_2\).

Our approach to drawing the Gordian graph involves brute force searching the space of all knot diagrams under, say, \(8\) crossings, systematically changing each crossing, and recording the resulting knot. Determining which knot we have as a result can be done by SnapPy, a program for low-dimensional topology capable of basic knot identification. Knot diagrams would likely be encoded by their planar diagram (PD) codes.

Having an interactive, searchable Gordian graph could be of enormous help to researchers looking for examples and counterexamples in low-dimensional topology to test their conjectures against. It may also help illuminate relationships between knots that are related by a single crossing change (for instance, every knot is at most one crossing change away from an Alexander polynomial \(1\) knot¹). Also, many results about the Gordian graph give existence results that are not constructive, meaning that the actual diagram and crossing that is changed is not given by the theorem.

Potential further project: surface-knot tabulation

One dimension up, embeddings of surfaces into \(S^4\), or knotted surfaces, can also be studied via their diagrams. One way to do so is using ch-diagrams, introduced by Yoshikawa². Ch-diagrams are similar to (classical) knot diagrams, but allow an additional type of crossing representing a saddle point in the surface.

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 (embeddings of \(S^1\) into \(S^3\)) 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 to ch-diagrams with more than \(10\) crossings. The strategy is similar to the Gordian graph project: first generate all diagrams with a certain number of crossings, filter out the diagrams which don’t produce surfaces, then determine which surfaces the diagrams correspond to.

Suggested Background

At least one of the following:

• Point-set topology. More advanced topology or familiarity with knot theory a plus but not required.
• Coding experience (Python preferred, since we will likely be using the knot theory software SnapPy)

Graphics/visualization experience for drawing the Gordian graph would be helpful but is not required.

Application Form

Here is the link to the application form.

References