Claremont Topology Seminar: Ryan Maguire (Dartmouth College)

We especially welcome all undergraduates and graduate students to attend topology seminar!

Speaker: Ryan Maguire (Dartmouth College)

Title: Relative Strengths of Knot Invariants by Experiment

Abstract: Four knot polynomials have been well studied by topologists, graph theorists, and algebraists alike:
The Alexander, Jones, HOMFLY-PT, and Khovanov polynomials. It is known that the Khovanov polynomial is “stronger” than the Jones polynomial, and similarly one may state that HOMFLY-PT is stronger than both the Alexander and Jones polynomials. No comparison can be made between the Jones and Alexander polynomials since there are families of knots with identical Alexander polynomials but distinct Jones polynomials, and vice-versa, but experiment tells us the Jones polynomial is stronger, on average, at distinguishing knots. We have tabulated the Alexander, Jones, and HOMFLY-PT polynomials for all knots up to 19 crossings, and the Khovanov polynomial for up to 17 crossings. Using this, we can experiment on the relative strengths of these knot invariants and generate statistics on them.