Papers
- “Skein Theory of Affine ADE Subfactor Planar Algebras” (in preparation)
- "A Determinant Formula of the Jones Polynomial for a Family of Braids" jt. with D. Asaner, S. Kumar, A. Pease, and A. Poudel (in preparation)
Research Overview
My advisor is Stephen Bigelow. My research interests are broadly in low-dimensional topology, quantum algebra, and operator algebras.
My research concerns planar algebras: families of vector spaces together with multi-linear maps that can be encoded by diagrams in the plane. More specifically, I study subfactor planar algebras which were introduced by Vaughan Jones as an axiomatization of the standard invariant of a subfactor. These planar algebras also encode two other invariants of the subfactors: the index and the principal graph. The Kuperberg Program asks to find diagrammatic presentations of all subfactor planar algebras. This program has been completed for index less than 4. For my thesis, I am specifically focusing on completing The Kuperberg Program for index 4.
I was also a graduate mentor for a summer 2022 REU project at the intersection of knot theory and graph theory. There is a pairing between some knots and graphs such that certain knot polynomials can be recovered from a corresponding graph polynomial. Our REU group obtained the Jones polynomial for a family of links arising from the closure of a braid of a particular form using a graph polynomial arising from their balanced overlaid Tait graphs. My mentees typed their results in a poster, which you can view here. This project was led by Sanjay Kumar, and is also joint with Derya Asaner, Andrew Pease, and Anup Poudel.
I love discussing my research, so if you would like any more details on either of the projects, please feel free to email me.