I am a topologist, specializing in low-dimensional topology, algebraic topology and combinatorial topology. As a grad student at LSU, I once showed up to see my future PhD advisor Rick Litherland give a talk in the topology seminar. One of the other topologists noticed me, the new guy in the room, and said "So, you're interested in toopology?" I replied "Well, I'm interested in both algebra and topology, and I'm not sure which direction I want to go." He replied "In that case you should definitely be a topologist, since you can do as much algebra doing topology as you would just doing algebra." How true those words turned out to be!

A knot is a simple closed curve in space, meaning it has no loose ends and does not intersect itself. In addition to knots, we have links which are several curves knotted together and tangles which have endpoints which are fixed in place. The basic question in knot theory is "Given two knots, how can we tell if they are knotted in the same way or in different ways?" The reasons to care about this are many and not necessarily obvious, including:

  • Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined in part by how they're knotted
  • Certain antibiotics work by blocking the action of molecules called topoisomerase which change how DNA is knotted; blocking the unknotting of the DNA stops the bacteria reproducing
  • Perhaps surprisingly, the mathematics of knots is relevant to the search for a theory of quantum gravity, a major unsolved problem in physics
  • Besides, knots are just fun!

To tell knots apart, we need knot invariants, quantities we can calculate from a knot which stay the same no matter how we move the knot around in space. The earliest example of a knot invariant is the linking number, discovered by Gauss himself. Other famous examples include the Alexander polynomial, the Jones polynomial, the HOMFLYpt polynomial (co-discovered by my colleague Jim Hoste) and more recently Khovanov Homology.

My research focuses on computable knot and link invariants derived from algebraic structures defined by knot and link diagrams. There are various different structures depending on whether the knot diagrams we're looking at are oriented, framed (i.e. kotted tori as opposed to knotted curves), both, or neither, as well as how we divide up the knot to get our algebraic generators; they have names like kei, quandles, racks, biquandles and biracks. See my quandles page for more.

In addition to discovering and exploring new invariants of classical knots, I'm also interested in combinatorial generalizations of knots such as virtual knots and higher-dimensional knots such as knotted surfaces in 4-dimensional space.

As an active researcher at an undergraduate institution, I do a lot of research with undergraduate student collaborators. These projects can be senior thesis projects, summer research (including REUs), independent study courses, or just plain doing a project to see where it goes. My goal is always a joint publication in a professional journal, and I have an extensive record of success. If you are a student who's interested in collaborating on a real research project (not just a term paper that gets labeled "research") in knot theory, drop by my office or send me an email!

Copyright © 2004-2010 Sam Nelson