Honors Oral Exam

Quandle invariants of knots and links.

Jacob Miller (University of Rochester)

Friday, April 29th, 2022
10:00 AM - 10:50 AM
Zoom Meeting ID: 986 2892 7517

Quandles are an algebraic structure whose axioms arise from the Reidemeister moves in knot theory. Given a knot \(K\) one can associate it with a quandle \(Q_{K}\) called the fundamental quandle of the knot. It was shown by David Joyce (1982) that if \(Q_{K}\) is isomorphic to \(Q_{K'}\) then the knots \(K\) and \(K'\) are equivalent up to orientation. Furthermore, Fenn and Rourke (1992) showed that the fundamental quandle is a complete invariant up to mirror image for non-split links. The fundamental quandle is very powerful in that all other invariants of classical knots can be viewed through their fundamental quandles. This survey provides a quick introduction to knot and quandle theory, gives examples of how we can view weaker invariants in-terms of the fundamental quandle through quandle-colorings , and describes stronger invariants derived from the fundamental quandle.

Event contact: jonathan dot pakianathan at rochester dot edu