Low-Dimensional Topology

 

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My research in low-dimensional topology is focused on the following projects:

 

  • Stability properties of the coefficients of quantum invariants.
  • Connection between q-series and the skein theory associated with various skein modules.
  • Invariants of Singular knots
  • Quandles and their applications to Knot Theory.

For my work on the colored Jones polynomial I often use Mathematica for testing and computation. I have written some scripts to compute the colored Jones polynomial on some knot families that I am making available here.

A list of my articles whose subjects are related to low-dimensional topology are given below.

The coefficients of the colored Jones polynomial

  • Twist Regions and Coefficients Stability of the Colored Jones Polynomial (Joint work with Mohamed Elhamdadi and Masahico Saito), Accepted to Transactions of the AMS 2017. Arxiv version.
  • The colored Kauffman skein relation and the head and tail of the colored Jones polynomial, Journal of Knot Theory and Its Ramifications 2017. Arxiv version.
  • The Tail of a Quantum Spin Network, The Ramanujan Journal, 2015. Arxiv version.
  • Pretzel Knots and q-Series (Joint work with Mohamed Elhamdadi), Osaka Journal of Mathematics, 2016. PDF.
  • The Bubble skein element and applications, Journal of Knot Theory and Its Ramifications, 2015. Arxiv version.

Singular knots invariants

The Jones polynomial and its generalizations 

  • On Rational Knots and Links in the Solid Torus, (Joint work with Khaled Bataineh, Mohamed Elhamdadi), submitted, 2017.
  • Jones polynomial for links in the handlebody (Joint work with Khaled Bataineh), Rocky Mountain Journal of Mathematics, Vol. 43, No. 2, 2013.