Research

Research Interests

• Low-dimensional Geometric Topology and Quantum Topology 

  • Kauffman bracket skein algebras

  • quantum invariants of links and 3-manifolds

  • topological quantum field theory

  • relationships between quantum topology, non-quantum topology, and hyperbolic geometry

• Applications of Topology

  • knotting in biopolymers

  • topological quantum computation

  • analysis of data using topology and geometry

Research Publications

Low-dimensional Geometric Topology and Quantum Topology

• (with Hiroaki Karuo and Han-Bom Moon) Cluster Algebras and Skein Algebras of Surfaces arXiv:2503.12319

• (with Hiroaki Karuo and Han-Bom Moon) Center of generalized skein algebras arXiv:2501.10686 

• (with Sike Wang) Fast algorithm for multiplication on skein algebra of one-hole torus. arXiv:2405:2059

• (with Francis Bonahon and Tian Yang) Asymptotics of invariants of surface diffeomorphisms II: the figure-eight knot complement. arXiv:2203.05730

• (with Francis Bonahon and Tian Yang) Asymptotics of invariants of surface diffeomorphisms I: conjecture and algebraic computations. arXiv:2112.12852

• (with Han-bom Moon) Consequences of the compatibility of skein algebra and cluster algebra on surfaces. New York Journal of Mathematics, Volume 30 (2024), 1648--1682.

• (with Han-bom Moon) On the Poisson Structure of the Roger-Yang Kauffman bracket arc algebra. Quantum Topology, Volume 12, Issue 2 (2021), 265-308.

• (with Francis Bonahon) Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality.  Quantum Topology,  Volume 1, Issue 2 (2019), 325-398.

• (with Francis Bonahon) Representations of the Kauffman bracket skein algebra II: Punctured surfaces. Algebraic & Geometric Topology, Volume 17, Number 6 (2017), 3399-3434.

• (with Martin Bobb, Stephen Kennedy and Dylan Peifer) The Kauffman bracket arc algebra is finitely generated.  Journal of Knot Theory and its Ramifications, Volume 25, Number 6 (2016), 1650034, 14pp.

• (with Martin Bobb, Stephen Kennedy and Dylan Peifer)  Presentations of Kauffman bracket arc algebra for small surfaces.   Involve: A Journal of Mathematics, Volume 9, Number 4 (2016), 689-698.

• (with Francis Bonahon) The Witten-Reshetikhin-Turaev representation of the Kauffman bracket skein algebra.  Proceedings of the American Mathematical Society, Volume 144, Number 6 (2016), 2711-2724.

• (with Francis Bonahon) Representations of the Kauffman bracket skein algebra I: Invariants and miraculous cancellations. Inventiones Mathematicae, Volume 204, Number 1 (2016), 195-243.

• (with Francis Bonahon) Quantum traces for representations of surface groups in SL2.  Geometry & Topology, Volume 15, Number 3 (2011), 1569-1615.

• (with Francis Bonahon) Kauffman brackets, character varieties, and triangulations of surfaces. Topology and geometry in dimension three (Proceedings of JacoFest), 179-194.   Contemporary Mathematics Series, Volume 560, American Mathematical Society, Providence, RI, 2011. 

• (with Nathan Dunfield) Quantum invariants of random Heegaard splittings.  Algebraic & Geometric Topology, Volume 11, Number 4 (2011),  2191-2205.

• Quantum invariants can provide sharp Heegaard genus bounds. Osaka Journal of Mathematics, Volume 48, Number 3 (2011), 709-718.

• SO(3) quantum invariants are dense in C. Mathematical Proceedings of the Cambridge Philosophical Society, Volume 148, Number 2 (2010), 289-295. 

• The SO(3) quantum invariants: their density and topological applications. Ph.D. thesis, Yale University, under the direction of Andrew Casson, 2007. 

Topology of Biopolymers

• (with Maciej Sikora, Erica Flapan, Pawel Rubach, Wojciech Gartska, Szymon Niewieczerzal, Eric Rawdon and Joanna Sulkowska) Proteins containing 6-crossing knot types and their folding pathways. bioRxiv:2023.06.16.545156

• (with Erica Flapan and Alireza Mashaghi) A tile modle of circuit topology for self-entangled biopoolymers. Scientific Reports, Volume 13, 8889 (2023).

• (with Erica Flapan and Adam He) Topological descriptions of protein folding. Proceedings of the National Academy of Sciences, Volume 116, Number 19 (2019), 9360--9369.

• (with Erica Flapan, Jeremy Grevet, Qi Li and Chen Daisy Sun) Knotted and linked products of recombination on T(2,n)#T(2,m) substrates. Journal of the Korean Mathematics Society, Volume 51, Number 4 (2014), 817--836.

Topological Quantum Computing

• (with Shawn Cui, Kevin Tian, Jennifer Franko Vasquez, Zhenghan Wang) The search for leakage-free, entangling Fibonacci braiding gates.  Journal of Physics A: Mathematical and Theoretical, Volume 52, Number 45 (2019) 455301

• (with Jennifer Franko Vasquez and Zhenghan Wang) Qubit representations of the braid groups from generalized Yang-Baxter matrices.  Quantum Information Processing, Volume 15, Number 7 (2016), 3035-3042.

Books and Edited Volumes

• (edited with Erica Flapan) Topology of Biopolymers. Contemporary Mathematics Volume 746, American Mathematical Society, 2020. 

• (with Erica Flapan and other members of the Park City Mathematics Institute Undergraduate Faculty Program) Knots, Molecules, and the Universe: An Introduction to Topology. American Mathematical Society, 2015. 

Selected Conference Presentations

• Autumn School on Low-dimensional Topology and Related Topics, IBS Center for Geometry and Physics, POSTECH, Korea, October 2024. 

• MAA Invited Paper Session: Knot Theory and Not Theory, MathFest, Indianapolis, August 2024.

• Quantum Topology, Quantum Information, and Connections to Mathematical Physics (in honor of Arthur Jaffe), Texas A&M, May 2024.

• AMS Special Session: Applied Topology, Joint Mathematics Meetings, Boston, January 2023. 

• Quantum Symmetries: Tensor Categories, Topological Quantum Field Theories, Vertex Algebras, Centre de Recherches Mathematiques, Universite de Montreal, October 2022. 

• From Subfactors to Quantum Topology (in memory of Vaughn Jones), University of Geneva, June 2022. 

• Quantum Topology and Geometry (in honor of Vladimir Turaev), Institut Henri Poincare, June 2022. 

• Noncommutative Algebras, Representation Theory, and Special Functions, Centre de Recherches Mathematiques, Universite de Montreal, June 2022. 

Conference Organizing

• (with Dorothy Buck, Danielle O'Donnol, and Biji Wong) A conference in honor of Erica Flapan, Summer 2026. 

• (with Aaron Lauda, Joshua Susan) Topological quantum computation, AMS Sectional specail session, Cal State San Luis Obispo, May 2025. 

• (with Bahar Acu, Ko Honda, Yi Ni) N+12th Southern California Topology Conference, March 2025. 

• (with Reginald Anderson) Quandles: A celebration of Sam Nelson's 100th paper, held virtually, May 2024. 

• (with Effie Kalfagianni, Christine Lee, and Tian Yang) Quantum Invariants and Low-Dimensional Topology, American Institute of Mathematics, August 2023. 

• (with Ko Honda, Aaron Lauda and Yi Ni) N+11th Southern California Topology Conference, held virtually, March 2021.

• (with Colleen Delaney and Jennifer Franko Vasquez) Braid groups and quantum computing, AWM Research Symposium special session, Rice University, April 2019. 

• (with Erica Flapan) Topology of biopolymers, AMS Sectional special session, Northeastern University, April 2018. 

• (with Charles Frohman) Topological representation theory, AMS Joint Meetings special session, Seattle, January 2016. 

• (with Michel Boileau, Louis Funar, Dave Futer, Francois Gueritaud, Ko Honda, Frederic Paulin) Classical and quantum hyperbolic geometry and topology: A conference in honor of Francis Bonahon, Universite Paris-Sud (Orsay), France, July 2015

• (with David Damiano and Scott Taylor) Topological, Geometric, and Quantum invariants of 3-manifolds, AMS Special Session,  College of the Holy Cross, April 2011.