I am a postdoctoral fellow in the Department of Mathematics at Colorado State University.
My research interests primarily lie within random topology and geometry through combinatorial and stat-mech methods, with a view towards knot theory. This research has connections through various lenses to biology, physics, and chemistry. I completed my PhD in May of 2017 under the direction of Jason Cantarella. Recently I have been studying knots from the map-theoretic view of knot diagrams, and my research interests include low-dimensional topology, differential geometry, combinatorics, and probability.
I love to write software for use in research, teaching, and even just for fun. This includes:
During Summer 2012 and the 2012-2013 school year, I was supported by the University of Georgia's VIGRE II grant. I graduated from Bowdoin College in May of 2011 with a B.A. in Mathematics and Computer Science, with Honors in Mathematics.
For the semester Fall 2017 at CSU, I'm teaching two sections of MATH 369 (Linear Algebra I).
When I was at UGA, I tought Calculus for Science and Engineering (Spring 2014, Spring 2016, Fall 2016), and Precalculus (Fall 2013, Fall 2015, Spring 2017), and I was a Writing Intensive Program teaching assistant for a robotics-based Calculus I course (Fall 2015).
Office: 223C Weber
Email: hchaps [at] gmail.com
We study random knotting by considering knot and link diagrams as decorated, (rooted) topological maps on spheres and pulling them uniformly from among sets of a given number of vertices n. This model is an exciting new model which captures both the random geometry of space curve models of knotting as well as the ease of computing invariants from diagrams. This model of random knotting is similar to those studied by Diao et al., and Dunfield et al. We prove that unknot diagrams are asymptotically exponentially rare, an analogue of Sumners and Whittington's result for self-avoiding walks. Our proof uses the same idea: We first show that knot diagrams obey a pattern theorem and exhibit fractal structure. We use a rejection sampling method to present experimental data showing that these asymptotic results occur quickly, and compare parallels to other models of random knots. We finish by providing a number of extensions to the diagram model. The diagram model can be used to study embedded graph theory, open knot theory, virtual knot theory, and even random knots of fixed type. In this latter scenario, we prove a result still unproven for other models of random knotting. We additionally discuss an alternative method for randomly sampling diagrams via a Markov chain Monte Carlo method.
We study random knotting by considering knot and link diagrams as decorated, (rooted) topological maps on spheres and pulling them uniformly from among sets of a given number of vertices n, as first established in recent work with Cantarella and Mastin. The knot diagram model is an exciting new model which captures both the random geometry of space curve models of knotting as well as the ease of computing invariants from diagrams. We prove that unknot diagrams are asymptotically exponentially rare, an analogue of Sumners and Whittington’s landmark result for self-avoiding polygons. Our proof uses the same key idea: we first show that knot diagrams obey a pattern theorem, which describes their fractal structure. We examine how quickly this behavior occurs in practice. As a consequence, almost all diagrams are asymmetric, simplifying sampling from this model. We conclude with experimental data on knotting in this model. This model of random knotting is similar to those studied by Diao et al, and Dunfield et al.
We consider a natural model of random knotting—choose a knot diagram at random from the finite set of diagrams with n crossings. We tabulate diagrams with 10 and fewer crossings and classify the diagrams by knot type, allowing us to compute exact probabilities for knots in this model. As expected, most diagrams with 10 and fewer crossings are unknots (about 78\% of the roughly 1.6 billion 10 crossing diagrams). For these crossing numbers, the unknot fraction is mostly explained by the prevalence of ‘tree-like’ diagrams which are unknots for any assignment of over/under information at crossings. The data shows a roughly linear relationship between the log of knot type probability and the log of the frequency rank of the knot type, analogous to Zipf’s law for word frequency. The complete tabulation and all knot frequencies are included as supplementary data.
For a full list, see my research page.