Foggy Bike Ride

Research Interests

I work in the field of Heegaard Floer theory, a subfield of topology that focuses on using Heegaard Floer homology to understand low-dimensional manifolds. At the moment, I am wrapping up a paper that develops a computational algorithm for Combinatorial Bordered Floer Homology with integral coefficients. Since there are currently only algorithms for coefficients over $\mathbb{Z}/2\mathbb{Z}$, I plan to start applying this algorithm to some common families of bordered manifolds, such as knot complements, to see what sort of results start appearing in the world of integral coefficients.

Due to the mostly combinatorial nature of the algorithm, this research would be suitable for collaboration with high-level undergraduate students looking for research opportunities.

Papers on the arXiv

  1. Knowles, Douglas and Petkova, Ina. ``Bordered Floer Homology with Integral Coefficients for Manifolds with Torus Boundary." ArXiV,
  2. Heap, Aaron, and Knowles, Douglas. "Space-efficient knot mosaics for prime knots with mosaic number 6." Involve, vol. 12, 2018.
  3. Heap, Aaron, and Knowles, Douglas. ``Tile Number and Space-Efficient Knot Mosaics." Journal of Knot Theory and Its Ramifications, vol.27, no.6, 2018.
  4. Coons, Jane Ivy, et al. "Numerical Ranges over Finite Fields." Linear Algebra and Its Applications , vol. 501, 2016, pp.37-47.