In my research, I investigate various measures of knot complexity. A knot is an embedded circle in 3 dimensions; one can think of a sailor's knot but with the ends of the rope glued together. Each non-trivial knot has crossings where one strand of the knot passes over another. The unknotting number of a knot measures the number of crossings that must be changed (cut open and twisted around each other once) in a knot in order to untie it. I study the effect of twisting an even number of strands of a knot under a "null-homologousness condition"; the minimal number of these twists needed to untie a knot is called the untwisting number, which is related to a beautiful equivalence relation on 3-manifolds due to Cochran, Gerges, and Orr and sparked a new theory of finite type invariants for 3-manifolds.
My research interests also include the theory of 3- and 4-manifolds, the theory of mechanical games such as Lights Out!, crossing numbers of graphs, topological data analysis, and scholarship on teaching & learning.
Selected papers & book chapters
The untwisting number of a knot (arXiv link; in the Pacific Journal of Mathematics, 2016)
Untwisting information from Heegaard Floer homology (arXiv link; in Algebraic & Geometric Topology, 2017)
(joint with Tara Davis, Lauren Grimley, Gizem Karaali, Boyan Kostadinov, and Roberto Soto) From Puzzles to Proofwriting: Exploring Rich Mathematical Ideas through Mechanical Puzzles (in Teaching Mathematics through Games, ed. Mindy Capaldi, MAA Press, 2021)
(joint with Chad Topaz, James Cart, Carrie Diaz Eaton, Anelise Hanson Shrout, Jude A. Higdon, Brian Katz, Drew Lewis, and Jessica Libertini) Demographic study of signatories to public letters on diversity in the mathematical sciences; in PLoS ONE, 2020; Science magazine coverage)
Unknotting via null-homologous twists and multi-twists (joint w/ Samantha Allen, Seungwon Kim, Benjamin Matthias Ruppik, and Hannah Turner; arXiv link; supported by 2022 MSRI SRiM, AIM 4-D Topology Research Community, & 2022 Gore Individual Summer Grant)
Textbooks in preparation
Quantitative Reasoning for Social Justice: an Active Learning Workbook (forthcoming; supported by a Westminster College Merit Leave)
Papers in preparation
(joint with Stefan Friedl) A note on the table theorem
Research with students
Kepuali Otuafi and Addison Scanlon. Controlling for police presence in Salt Lake City police-public interactions (forthcoming Joint Mathematics Meetings poster presentation 2022)
Aria Cederlof. Racial and gender analysis of the use of force and street checks by the Salt Lake City Police Department (2017) [preliminary draft; talk at Roanoke College]
Known untwisting numbers for knots up to 12 crossings
I have compiled a spreadsheet of all known untwisting numbers (see [Ince, 2017] for definitions and bounds) for knots up to 12 crossings, as well as the values of lower and upper bounds on the untwisting number and a couple of other relevant invariants. The lower bounds of topological 4-genus and algebraic unknotting number are highlighted in green, and the upper bounds of twice the 3-genus and the standard unknotting number are highlighted in blue. If you've determined the untwisting number of a knot in this table and I haven't filled it in yet, I'd very much appreciate it if you contacted me (kince at westminstercollege dot edu).
Others' work on the untwisting number (incomplete list)
Celoria, Daniele, and Marco Golla. (2019). Heegaard Floer Homology and Concordance Bounds on the Thurston Norm. Transactions of the American Mathematical Society (2019): 1.
McCoy, Duncan. (2019). Gaps between consecutive untwisting numbers.
Baader, Sebastian., Banfield, Ian., & Lukas Lewark. (2019). Untwisting 3-strand torus knots.
Allen, S., & Livingston, C. (2020). Unknotting with a single twist.