Research

I am interested in low dimensional topology, symplectic topology, contact topology, and knot theory. In particular, three-dimensional contact topology and knot homologies. You can find my research statement from 2019 here.

A (topological) knot is an embedding of S1 into  ℝ3, but can be thought of as a string tangled up, which then has the ends  glued together. The unit circle in the xy-plane is an example of a topological knot, called the unknot. One goal in studying knots, is to classify them. We do this using knot invariants, algorithms for assigning a number, group, an algebra, etc. to a knot such that if two knots are isotopic (the "same"), they are assigned the same value.

As with topology, where adding geometric structure to manifolds can yield further information about the topological manifold, studying Legendrian knots (knots which satisfy additional geometric constraints) can yield further information about topological knots. The classification of Legendrian knots is a much finer classification than that of the topological knots; the topological knot type of a Legendrian knot (considering the knot without the extra geometric constraints) is a Legendrian knot invariant. In fact, there are infinitely many non-isotopic Legendrian knots that are topologically the unknot (isotopic to the circle). 

One way to study Legendrian knots in  ℝ3 with the standard contact structure is to consider their projection to the xz-plane. The boundaries of the colored regions in the figure above is an example of a projection to the xz-plane of a Legendrian left-handed trefoil. A Legendrian knot invariant that comes from studying these projections are normal rulings. A projection of a Legendrian knot to the xz-plane has a normal ruling if you can decompose the diagram into circles which bound topological disks such that each circle has exactly one left cusp and one right cusp and the circles only intersect in certain ways. The figure above is an example of an xz-projection of a Legendrian left handed trefoil knot in plat position with a normal ruling indicated. The boundary of each colored (topological) disk is one of the circles.

For more information on what all of these terms mean, see my paper Augmentations and Rulings of Legendrian Knots.

Upcoming Conferences

• Representation Theory, Symplectic Geometry, and Cluster Algebras at BIRS, Spring 2025

Papers

• Lagrangian Realizations of Ribbon Cobordisms with John Etnyre. (arXiv)

• Lagrangian Cobordism of Positroid Links with Johan Asplund, Youngjin Bae, Orsola Capovilla-Searle, Marco Castronovo, and Angela Wu. (arXiv) Pacific Journal of Mathematics 332 (2024), no. 1, 1-21.

• Constructions of Lagrangian Cobordisms with Sarah Blackwell, Noémie Legout, Maÿlis Limouzineau, Ziva Myer, Yu Pan, Samantha Pezzimenti, Lara Simone Suárez, and Lisa Traynor. (arXiv). Chapter in: Acu, B., Cannizzo, C., McDuff, D., Myer, Z., Pan, Y., Traynor, L. (eds) Research Directions in Symplectic and Contact Geometry and Topology. Association for Women in Mathematics Series, vol 27 (2021). Springer, Cham.

• Satellite Ruling Polynomials, DGA Representations, and the Colored HOMFLY-PT Polynomial with Dan Rutherford. (arXiv). Quantum Topol. 11 (2020), 55-118. (Video from a talk below.)

• Augmentations and Rulings of Legendrian Links in #^k(S^1\times S^2)  (arxiv), Pacific Journal of Mathematics 288 (2017), no. 2, 381-423.

• Augmentations and Rulings of Legendrian Knots (arXiv), Journal of Symplectic Geometry Vol. 14 (2016), no. 4, 1089-1143. (Slides below.)

• Augmentations and Rulings of Legendrian Links. Diss. Duke University, 2016. (thesis)

• Semi-local formal fibers of minimal prime ideals of excellent reduced local rings, Journal of Commutative Algebra 4 (2012), no. 1, 29-56. Joint with N. Arnosti, R. Karpman, J. Levinson, and S. Loepp.

Research with Undergraduates

• See Bard Digital Commons for theses from the following students' senior projects:

  • 2022-2023 Max Redman (joint with CS, co-advised with Bob McGrail. Quandle Unification Reduces to Quandle Matching

  • 2021-2022 Ansel Tessier (joint with CS, co-advised with Bob McGrail. Does Bias Have Shape? An Examination of the Feasibility of Algorithmic Detection of Unfair Bias Using Topological Data Analysis

  • 2020-2021  Raphael Walker. Lagrangian Cobordisms of Legendrian Pretzel Knots with Maximal Thurston-Bennequin Number

• Here is a poster by DeVon Ingram and Hunter Vallejos from their REU at Georgia Tech Summer 2018.

• Here is a poster by Hunter Vallejos from his REU at Georgia Tech Summer 2017.

Slides, posters, and videos

• Video from talk on my paper with Dan Rutherford, at the Simons Center Workshop on Categorification in Mathematical Physics here.

• Topology Students Workshop (June 2014) slides from talk on my paper Augmentations and Rulings of Legendrian Knots are here.

• Poster on the same paper is here.

• Duke Math Slam (March 2014) slides from my short talk on knots and 3-manifolds are here.