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

Textbooks in preparation

Papers in preparation

Research with students

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)