Research

Here is a list of my publications and preprints. The years indicate when the results in the papers were proven.

  1. 2020: Lectures in knot theory: An Exploration of Contemporary Topics, Springer Universitext (in preparation, accepted for publication), with J. H. Przytycki, D. Ibarra, G. Montoya-Vega, and D. Weeks. 
  2. 2020: Kauffman bracket skein module of the connected sum of handlebodies: A counterexample, with J. H. Przytycki, Manuscripta Math. (accepted), e-print: arXiv:2005.07750.
  3. 2020: On framings of links in 3-manifolds, with D. Ibarra, J. H. Przytycki, G. Montoya-Vega, D. Weeks, e-print: arXiv:2001.07782 (submitted).
  4. 2019: A Generalization of the Gram determinant of type A, with S. Mukherjee, D. Ibarra, J. H. Przytycki, e-print: arXiv:1905.07834 (submitted).
  5.  2018: Schur multipliers and second quandle homology, with S. Mukherjee D. Ibarra, T. Nosaka and J. H. Przytycki, J. Algebra, DOI: 10.1016/j.jalgebra.2019.12.027, e-print: arXiv:1812.04704.
  6. 2018: On multiplying curves in the Kauffman bracket skein algebra of the thickened four–holed sphere, with S. Mukherjee, J. H. Przytycki, M. Silvero and X. Wang, J. Knot Theory Ramifications (accepted), e-print: arXiv:1805.06062.

Here are the slides from a few select talks that I have delivered.

Skein Modules Through The Ages (Knots in Washington XLIX, February, 2020)

Abstract: This talk will give the audience a tour of the history and development of skein modules of 3 – manifolds and their evolution into one of the most important tools in knot theory having connections to hyperbolic geometry, the SL(2,C) character variety and the Witten – Reshetikhin – Turaev topological quantum field theory. We will end the presentation with a discussion of Witten’s finiteness conjecture for Kauffman bracket skein modules.

Gram Determinants Motivated By Knot Theory (Student Research Talks (StReeTs), George Mason University, October 2019)

Abstract: The study of Gram determinants in knot theory dates back to Lickorish when he used Gram matrices arising from the Temperley – Lieb algebra and constructed the Witten-Reshetikhin-Turaev invariant of 3 – manifolds combinatorially. In this talk, I will first discuss Gram determinants of type A and Gram determinants of type B. A closed formula for the latter was given by Chen and Przytycki which answered a question posed by the late combinatorialist Rodica Simion about the relationship between matrices of chromatic joins and Gram type matrices. Finally, I will describe our work on the Gram determinant of generalised type A and the Gram determinant of type Mb.