Research

pretzel
From left to right: a pretzel knot with 2 positive twists in the first tangle and 5 negative twists in the second and third tangle, a Conway sphere decomposition, and the corresponding curve systems on the 4-punctured spheres from the decomposition.

I am a low-dimensional topologist interested in knot theory. In particular, I would like to understand the connection between different knot invariants, such as the Jones-types invariants motivated by representation theory and physics, and the hyperbolic geometry of knot complement. One way to do this is to relate them through the diagram of a knot. Much of my research has concerned the relationship of the colored Jones polynomial, a quantum invariant which contains the Jones polynomial, to the diagram and the geometry of the knot complement. Recently, I have also become interested in knot and 3-manifold invariants coming from Heegaard Floer Homology, contact geometry, and normal surface theory.


The material below is based on research partially supported by the National Science Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

Papers

The versions on the arXiv may differ from the published versions.
  1. Jones slopes and coarse volume for near-alternating links. arXiv
  2. Normal and Jones surfaces of knots, with Efstratia Kalfagianni, submitted. arXiv
  3. A trivial tail homology for non A-adequate links, accepted for publication in Algebraic and Geometric Topology. arXiv
  4. Slopes for pretzel knots, with Roland van der Veen, New York Journal of Mathematics, 22, p. 1339--1364, 2016. arXiv, published
  5. Stability properties of the colored Jones polynomial, submitted. arXiv
  6. Crosscap numbers and the Jones polynomial, with Efstratia Kalfagianni, Advances in Mathematics, 286, 308-337, 2016. arXiv, published
  7. On the degree of the colored Jones polynomial, with Efstratia Kalfagianni, Acta Mathematica Vietnamica. (Proceedings of Quantum Topology and Hyperbolic Geometry in Nha Trang, May 2013), 39(4), 2014. arXiv, published

Preprints and Notes

Talks

  1. The colored Jones polynomial and surfaces in 3-manifolds, Sep 28th, 2017, Séminaire Topologies, Institut Montpelliérain Alexander Grothendieck.
  2. The colored Jones polynomial and surfaces in 3-manifolds, Aug 21st, 2017, Oberwolfach Workshop in Low-dimensional Topology and Number Theory, Oberwolfach, Germany.
  3. Jones slopes and coarse volume for near-alternating links, May 6th, 2017, Special session on Invariants in Low-dimensional Topology, Spring Eastern Sectional Meeting, New York, NY.
  4. Jones slopes and Murasugi sums of links, March 22nd, 2017, Virtual Seminar, Lousiana State University, Baton Rouge, LA.
  5. Jones slopes and Murasugi sums of links, February 20th, 2017, Geometry and Topology Seminar, Georgia Institute of Technology, Atlanta, CA.
  6. Jones slopes and Murasugi sums of links, February 9th, 2017, Topology Seminar, Michigan State University, East Lansing, MI.
  7. Slopes for near-adequate knots, January 6th, 2017, Special session on Women in Topology, Joint Mathematics Meeting, Atlanta, GA.
  8. A trivial tail homology for non A-adequate links, December 4th, 2016, Session on Geometric Group Theory and Topology in Low Dimensions, CMS Winter Meeting, Niagara Falls, ON, Canada.
  9. Stability in the categorification of the colored Jones polynomial, October 9th, 2016, Special session on Floer theoretic invariants of 3-manifolds and knots, University of Denver, Denver, CO.
  10. Quantum knot invariants and 3-manifolds, August 9th, 2016, Workshop on knot theory and related topics, RIMS, Kyoto, Japan.
  11. The colored Jones polynomial and boundary slopes of pretzel knots, July 19th, 2016, Knots in Hellas, Ancient Olympia, Greece.
  12. Slope Conjectures for Pretzel Knots, Topology Seminar, California Institute of Technology, Pasadena, CA.
  13. The colored Jones polynomial and Slopes for Pretzel Knots, Topology Seminar, Rice University, Houston, TX.
  14. Link Homology from Bar-Natan Skein Modules, January 6th, 2016, Joint Mathematics Meetings Special Session: Topological Representation Theory, II. Seattle, WA.
  15. The Slope Conjecture and 3-string pretzel knots, October 3rd, 2015, AMS Central Fall Sectional Meeting Special Session: Geometric Perspectives in Knot Theory, II. Loyola University Chicago, Chicago, IL.
  16. The colored Jones polynomial of non semi-adequate links. July 21st, 2015, First Encounter to Quantum Topology: School and Workshop, KIAS, Seoul, Korea.
  17. The Slope Conjecture and 3-string pretzel knots. April 3rd, 2015, Spring 2015 Redbud conference, Oklahoma State University, Stillwater, OK.
  18. Crosscap number of an alternating link and the Jones polynomial, March 15th, 2015, AMS Special Session on Geometry and Invariants of 3-Manifolds, Michigan State University, East Lansing, MI.
  19. The Jones polynomial, 3-braids, and L-space knots, January 8th, 2015, Knot Theory and its Applications to Physics and Quantum Computing, University of Texas at Dallas, Richardson, TX.
  20. A stability result on the colored Jones polynomial, October 21st, 2014, University of Iowa, Iowa City, IA.
  21. Crosscap Number and the Jones polynomial, The Thin Manifold: A conference and graduate student workshop on knots and 3-manifolds. August 7th, 2014, University of Iowa, IA.
  22. Stability Properties of the Colored Jones polynomial, Twelfth Annual Graduate Student Topology & Geometry Conference, April 5th, 2014, University of Texas at Austin, Austin, TX.