This project was comentored with Kevin Santos (which at the moment of writing this is a graduate student in U of T).
What was the question?
Find a criterion to determine when a knot is “divisible” by a trefoil.
What motivated me?
A famous result in knot theory is that every knot can be decomposed uniquely into prime knots. The first prime knot, beyond the trivial one, is the trefoil. I am a number theorist, and at the point of thinking on this, I was teaching elementary number theory, and one topic is precisely divisibility. There is a criterion for when a number is divisible by 3, and this reminded me of the trefoil and the above theorem. I thought that we could find some criterions.
What did the students do?
Studied many different types of invariants and tried to connect them with their relevance with being a multiple of the trefoil. There were three types of invariants they studied: polynomial invariants as the Conway and Alexander Polynomials. They also studied Fox colorings and what being divisible by the trefoil implies about the number of those coloring that the knot can have. Finally, they studied properties associated to the Knot Group and which kind of maps exists from it to the permutation group S_3 when the knot is divisible by the trefoil.
We learned a lot of knot theory and we found out that all of the above results gave the same answers. They were able to distinguish the same results one from the others. The students showed that in certain type of knots (the torus knots) the behavior follow certain patterns related to the Conway Polynomial constant term.
Special moments I remember 🙂
I really liked when the students working together on different paths finally realized that the criteria were somewhat equivalent. The process they had to do to see if this was known in the literature (and it was but in very disconnected way) was very interesting.
Outcomes of the work
The students competed for a place at the CUMC 2024. They were not selected but the work was quite successful for them.
Malors 