Research Statement Multiplication Tangle

In my thesis, I use the planar algebra defined by Bigelow [4] to describe the multivariate Alexander polynomial. The advantage to using a planar algebra in this way is that the multivariate Alexander polynomial generalizes to a multivariate tangle invariant up to Reidemeister I. The planar algebra that I used in my thesis is the Motzkin planar algebra, which has connections to the representation theory of Lie algebras [3] that I study now. In Section 1 of my research statement, I give some background on unshaded planar algebras and discuss the classical example of the Temperley-Lieb planar algebra. In Section 2, I discuss the main result of my thesis and its relationship to subfactor theory. I describe my current interest in the representation theory of Lie algebras and how it relates to my work with the multivariate Alexander polynomial in Section 3.