UConn Math Club

Using Polynomials to Study Knots

Adam Giambrone (University of Connecticut)

Wednesday, January 31, 2018 5:45 pm
MONT 321

Knot theory is the study of knotted loops in space. This field, originally developed out of a failed attempt in the late 1800s to make a periodic table of elements, now has connections to many other areas of mathematics and science. A key strategy to tell knots apart from each other is to use polynomials associated to knots in such a way that deforming the knot in space has no effect on the polynomial (equivalent knots have equal polynomials).

In this talk, we will explore how to use knot invariants to study knots up to equivalence. In particular, we will focus on a powerful knot polynomial called the Jones polynomial. For instance, the knot below on the left has Jones polynomial $t^2-t 2-2t^{-1} t^{-2}-t^{-3} t^{-4}$ (we allow negative exponents in our polynomial) and the knot below on the right has Jones polynomial $-t^3 2t^2-2t 3-2t^{-1} 2t^{-2}-t^{-3}$ (a different polynomial), which tells us that even with an infinite amount of time, we would never be able to deform the knot on the left to be the knot on the right. What a time saver!

Comments: Free pizza and drinks!