Research

Presenting at the Innovative Pedagogy in Geometry and Topology Workshop at Oberlin College in July 2023.

Mathematical Interests:

Some of my research work is at the intersection of hyperbolic geometry and low-dimensional topology. I am interested in geometric and topological invariants of hyperbolic 3-manifolds, especially hyperbolic knot and link complements. My thesis and early research work focused on analyzing symmetries, hidden symmetries, and commensurability classes of these manifolds, along with connections to their length spectra. My thesis advisor was Dave Futer. My more recent work in geometric topology has involved a series of projects with Rollie Trapp (CSUSB) focused on exploiting the hyperbolic structures of fully augmented link complements to analyze the topology of these links and their respective complements .

I have also worked with undergraduate students on a variety of mathematical research projects, several of which have been in combinatorial topology. Several of these projects involved analyzing embeddings of Cartesian products of graphs on surfaces.

Finally, I have started working on projects at the intersection of historical cryptology, mathematics, and data analysis. I am interested in developing and improving algorithms and quantitative tools to assist with the cryptanalysis of classical ciphers, such as the Vigenere cipher, the Playfair cipher, and the Hill Cipher.

A copy of my CV can be found here.

The * symbol denotes an undergraduate student research collaborator.

Preprints and Works in Progress:

Embeddings of Repeated Cartesian Products of Paths and Cycles. (In Progress)

Joint with Y. Yau

Detecting the Block Size of the Hill Cipher. (In Progress)

Joint with Y. Yau

Projective Planar Cartesian Products of Graphs. (Submitted)

Joint with N. Abell* and E. McDermott*

Symmetry Groups of Flat Fully Augmented Links and Their Complements. (Submitted)

Joint with R. Trapp

Publications:

Using Integer Programming to Solve Games, Puzzles, and Ciphers.

Joint with E. Bouzarth, J. Harris, and K, Hutson, accepted in Foundations for Undergraduate Research in Mathematics (Springer).

Toroidal Cartesian Products Where One Factor is 3-Connected.

Joint with E. Badgett*, published in Graphs & Combinatorics, Vol. 42, 40 (2026). https://doi.org/10.1007/s00373-026-03044-6

Joint with L. Bouzarth, J. Harris, and K. Hutson

Symmetry Groups of Hyperbolic Links and Their Complements.

Joint with R. Trapp, published in The Journal of Knot Theory and Its Ramifications 34-8 (2025), Paper 2550033, 17 pages. DOI 10.1142/S0218216525500336, arXiv: 2403.17616

Flat Fully Augmented Links are Determined by Their Complements.

Joint with R. Trapp, published in Algebraic & Geometric Topology 25-8 (2025), 4839--4896. DOI 10.2140/agt.2025.25.4839.

An artificial neural network approach to finding the key length of the Vigenere cipher,

Joint with Y. Yau, published in Cryptologia Vol. 49 (4), 331-347 (2025). https://doi-org.libproxy.furman.edu/10.1080/01611194.2024.2351117

Modifying twist algorithms for determining the key length of the Vigenere cipher.

Joint with M. Carns*, A. Pate*, and Y. Yau, published in Cryptologia Vol. 48 (6), 558-573 (2024). https://www.tandfonline.com/doi/full/10.1080/01611194.2023.2275583

Deciphering Secrets Throughout History: An Interdisciplinary Linguistics and Cryptology Course.

Joint with E. Kim, published in the 6th International Conference on Historical Cryptology HistoCrypt 2023.

Symmetries and Hidden Symmetries of (\epsilon, d_L)-twisted knot complements.

Joint with N. Hoffman and W. Worden, published in Algebraic & Geometric Topology, 22-2 (2022), 601-656. https://msp.org/agt/2022/22-2/p03.xhtml

Embedding Grid Graphs on Surfaces.

Joint with Fabian Salinas*, published in Graphs and Combinatorics, Vol. 38, 87 (2022). https://doi.org/10.1007/s00373-022-02488-w

Dehn Surgery and Hyperbolic Knot Complements Without Hidden Symmetries

Joint with E. Chesebro, J. DeBlois, N. Hoffman, P. Mondal, and W. Worden, published in International Mathematics Research Notices (2022). rnac018, https://doi.org/10.1093/imrn/rnac018

Arithmeticity and Hidden Symmetries of Fully Augmented Pretzel Link Complements.

Joint with J. Meyer and R. Trapp, published in New York J. Math., Vol. 26 (2020), 149-183.

Mutations and Short Geodesics in Hyperbolic 3-Manifolds.

Published in Communications in Analysis and Geometry, Vol. 25, No. 3 (2017), 625-683.

Spectrally Similar Incommensurable 3-Manifolds.

Joint with D. Futer, published in Proceedings of the London Mathematical Society, Vol. 115, No. 2 (2017), 411-447.

Hidden Symmetries and Commensurability of 2-Bridge Link Complements.

Joint with W. Worden, published in Pacific Journal of Mathematics, Vol. 285, No. 2 (2016), 453-484.

Factorial Growth Rates for the Number of Hyperbolic 3-Manifolds of a Given Volume.

Published in The Proceedings of the American Mathematical Society, Vol. 143, No. 5 (2015) 2201-2214.

Math Horizons Publications

Crypto Corner. Math Horizons 33 (2025), no. 2, 2--2.
The Math Behind College Admissions. Math Horizons 22 (2014), no. 1, 14--15.

Joint with J. Cousins

Book Reviews

Review of Do Plants Know Math? Unwinding the Story of Plant Spirals from Leonardo da Vinci to Now. Math Horizons, Vol. 32, No. 7 (2025).
Review of the expanded edition of Codebreaking: A Practical Guide by Elonka Dunin and Klaus Schmeh. Cryptologia, 49(1), 94–96 (2024).

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