Research & Presentations

There's an old joke about how to visualize 4 dimensions: you picture something in 3-dimensions and say to yourself "4" very seriously.  The interesting thing is that, when done correctly, this essentially works!  

My research primarily involves studying knotted surfaces in 4-dimensions from a knot theoretic perspective.  More details can be found in my research prospectus.  

Preprints

"Moves on Bridge Multisections" (joint work with Román Aranda)

https://arxiv.org/abs/2410.01921

"Studying Knotted Surfaces as Loops in CW Complexes" (joint work with Román Aranda)

Read the first three sections of a preliminary draft     

"Generating Infinitely Many Hyperbolic Knots with Plats" (joint work with Seth Hovland)

https://arxiv.org/abs/2410.17443

Ongoing Projects

Towards Polynomial Unknotting

For the past year or so, I have been involved in a project lead by William Menasco and Greg Vinal which aims to develop a polynomial-time algorithm to recognize the unknot.  This is a large project with many graduate students involved.  Motivated by the work of Ivan Dynnikov, we have taken a variety of approaches in this problem; contact geometry and foliations on disks are the primary tools.  

A new L-type Invariant

In an ongoing collaboration with Román Aranda and Nobutaka Asano, we are developing an L-type invariant for knotted surfaces defined in the shadow diagram complex.  This project poses some interesting challenges, as there are not direct analogs for the homological arguments which Nobutaka Asano and his coauthors use in their paper "Some lower bounds for the Kirby-Thompson Invariant."  We've made progress using mostly geometric techniques, and intend to continue investigating what else shadow diagrams may be able to tell us about knotted surfaces. 

Plats, Shadows, and the Braid Group 

In my collaboration with Seth Hovland, we have discovered some interesting connections between shadow diagrams, the curve complex, and the homology of the three ball with a tangle removed.  We are investigating how the braid group can be applied to these connections, and whether this can lead to new methods of studying knots as plats.  In particular, we're using the shadow diagram complex to further study the Hilden double coset of plats.  This is an ongoing collaboration with many goals, but has been put on pause while both Seth and I are actively applying for jobs.  Read more about his awesome work here!

Presentations 

Representing Knotted Surfaces as Loops in CW Complexes

A Brief Introduction to Handlebodies (expository)

Three Ways to Visualize Knotted Surfaces (expository)