I mostly like to think about knots in 3-manifolds. Currently, I'm trying to understand when the fundamental group of the complement of a knot in S3 is bi-orderable, i.e. has a total order invariant under both left and right multiplication. Some examples of knots with bi-orderable knot groups are the figure 8 knot, the stevedore knot, and the pretzel knot P(-3,3,3). Typically, knot groups are not bi-orderable so naturally one wonders, "When a knot does have a bi-orderable knot group what does this tell us about the topology of the knot?" This is the motivating question behind my research.
Publications and Works in Progress
♦ Residual torsion-free nilpotence, bi-orderability, and two-bridge links, Canadian Journal of Mathematics (2023)
♦ Residual torsion-free nilpotence, bi-orderability and pretzel knots, Algebraic and Geometric Topology 24, 4 (2023)
♦ Non-standard bi-orders on punctured torus bundles, preprint, w/ H. Segerman
♦ Algorithmic obstructions and order-preserving braids, preprint, w/ N. Scherich and H. Turner
♦ Bi-orderability and double twist links, in preparation