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
- Generating sets of Reidemeister moves of oriented singular links and quandles, (Joint work with Khaled Bataineh, Mohamed Elhamdadi, William Youmans), submitted, 2017.
- Foundations of the Colored Jones Polynomial of singular knots, (Joint work with Mohamed Elhamdadi), Accepted to Bulletin of the Korean Mathematical Society, 2017. Arxiv version.
- Singular Knots and Involutive Quandles, (Joint work with Indu RU Churchill, Mohamed Elhamdadi, Sam Nelson) Accepted to Journal of Knot Theory and its Ramifications, 2017.
- The colored Jones polynomial of singular knots (Joint work with Mohamed Elhamdadi), New York J. Math 2016. Arxiv version.
The Jones polynomial and its generalizations