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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!
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