Research

I like thinking about knots, braids, surfaces, and 3-manifolds. My work to date has centered on:

• Homological invariants for knots and braids (Khovanov homology, annular Khovanov homology, knot Floer homology, etc.)

• Different perspectives on the braid group (algebraic, geometric, and topological), and their interactions

• Mapping class groups

• Contact topology, in particular the study of transverse knots

I also like thinking about possible applications of topology to biology and other fields. My research will be supported by an NSF LEAPS-MPS grant, award #2213451, from 2022-2024.

Preprints

• The Dehn twist coefficient for big and small mapping class groups, with Peter Feller and Hannah Turner (arXiv).

Papers

• On the Upsilon invariant of fibered knots and right-veering open books, with Dongtai He and Linh Truong  (arXiv), to appear in Mathematical Research Letters.

• Examples of non-minimal open books with high fractional Dehn twist coefficient, with Peter Feller (arXiv), published in the New York Journal of Mathematics, 917, 2022.

• Braids, fibered knots, and concordance questions, with Keiko Kawamuro, Feride Ceren Kose, Gage Martin, Olga Plamenevskaya, Katherine Raoux, Linh Truong, and Hannah Turner (arXiv). Published in the Springer volume Research Directions in Symplectic and Contact Geometry and Topology, 2021.

• Stability and triviality of the transverse element in Khovanov homology, with Christine Lee (arXiv),  published in Topology and its Applications, 107146, 2020.

• Braids with as many full twists as strands realize the braid index, with Peter Feller (arXiv),  published in the Journal of Topology, 12 (4), 1069-1092, 2019.

• A topological characterization of the products arising from site-specific recombination on T(2,n)#C(2,r) DNA substrates, with Jordan Katz (PDF). Published in the Journal of Mathematical Chemistry, 56 (6), 1728-1743, 2018.

• On sutured Khovanov homology and axis-preserving mutations (arXiv, published in the Journal of Knot Theory and its Ramifications, Volume 26 Number 4, 2017).

• An annular refinement of the transverse element in Khovanov homology, with Adam Saltz (arXiv, published in Algebraic & Geometric Topology, Volume 16 Number 4, 2016).

Mentoring

At Brooklyn College I have mentored independent study projects with Liam Gilmore-Greiss in Spring 2020  (introductory topics in topology) and Nachama Stern in Spring 2021 (topics included an introduction to graph theory, the classification of surfaces, and the curve graph).

At the University of Michigan, in Winter 2017, I co-mentored Amanda Burcroff, Wenyu Jin, and Max Kontorovich in a research project about billiards. My co-mentors were Sandrine Daurat and Jasmine Powell. In Winter 2018, I co-mentored Jason Liu, Juncheng An, Yi Zhou, and Yingsi Jian in a project through LOG(M) about triangulations of 3-manifolds with Dohyeong Kim and Mark Greenfield. See the Laboratory of Geometry at Michigan's website (LOG(M) for short) for more information and to see their final posters. 

In Summer 2017 I mentored Jordan Katz, an undergraduate at the University of Michigan, in a research project about applications of knot theory to DNA as part of Michigan's REU program. We wrote up the results which were accepted for publication in the Journal of Mathematical Chemistry (listed in the papers above).