Department of Mathematics

Colorado State University

Current Research

My research interests tend towards the intersection of combinatorics, topology and geometry, with a primary focus on random knots. I completed my PhD under the direction of Jason Cantarella.

A random alternating knot
A random alternating knot. I have shown that alternating knots are exponentially rare in the set of all knots.
A random alternating knot
A random trefoil knot diagram. It is believed that the knotted core of the diagram is small in relation to that of the whole diagram.

Publications

Entries are also available in BibTeX format.

  • On the structure and scarcity of alternating knots.
    Preprint: arXiv:1804.09780

    Given a class of objects, a pattern theorem is a powerful result describing their structure. We show that alternating knots exhibit a pattern theorem, and use this result to prove a long-standing conjecture that alternating knots grow rare. This is currently the best possible analogue of a pair of theorems on alternating links of Sundberg and Thistlethwaite in 1998 and Thistlethwaite in 1998, given the current obstructions to an exact enumeration of knot diagrams. We also discuss implications of this pattern theorem for subknots and slipknots in minimal alternating knot diagrams and types, partially answering a conjecture of Millett and Jablan.

  • A Markov chain sampler for plane curves.
    With Andrew Rechnitzer.
    Preprint: arXiv:1804.03311

    A plane curve is a knot diagram in which each crossing is replaced by a 4-valent vertex, and so are dual to a subset of planar quadrangulations. The aim of this paper is to introduce a new tool for sampling diagrams via sampling of plane curves. At present the most efficient method for sampling diagrams is rejection sampling, however that method is inefficient at even modest sizes. We introduce Markov chains that sample from the space of plane curves using local moves based on Reidemeister moves. By then mapping vertices on those curves to crossings we produce random knot diagrams. Combining this chain with flat histogram methods we achieve an efficient sampler of plane curves and knot diagrams. By analysing data from this chain we are able to estimate the number of knot diagrams of a given size and also compute knotting probabilities and so investigate their asymptotic behaviour.

  • Slipknotting in random diagrams.
    Preprint: arXiv:1803.07114

    The presence of slipknots in configurations of proteins and DNA has been shown to affect their functionality, or alter it entirely. Historically, polymers are modeled as polygonal chains in space. As an alternative to space curves, we provide a framework for working with subknots inside of knot diagrams via knotoid diagrams. We prove using a pattern theorem for knot diagrams that not only are almost all knot diagrams slipknotted, almost all unknot diagrams are slipknotted. This proves in the random diagram model a conjecture yet unproven in random space curve models. We also discuss conjectures on the enumeration of knotoid diagrams.

  • A diagrammatic theory of random knots.
    PhD thesis, University of Georgia.
    Preprint available here

    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.

  • Asymptotic laws for random knot diagrams.
    Journal of Physics A: Mathematical and Theoretical 50 (2017), no. 22, p. 225001.
    DOI: 10.1088/1751-8121/aa6e45
    Preprint: arXiv:1608.02638

    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.

  • Knot probabilities in random diagrams.
    With Jason Cantarella and Matt Mastin.
    Journal of Physics A: Mathematical and Theoretical 49 (2016), no. 40, p. 405001.
    DOI: 10.1088/1751-8113/49/40/405001
    Preprint: arXiv:1512.05749

    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.

Talks

External

  • Steady states for crossing changes in knot diagrams.
    The Topology of Nucleic Acids: Research at the Interface of Low-Dimensional Topology, Polymer Physics and Molecular Biology.
    Banff International Research Station (BIRS), Banff, AB, Canada, March 2019.
    Video available here.
  • Alternating knots are rare.
    Geometry, Topology and Dynamics Seminar.
    Okinawa Institute of Science and Technology, Okinawa, Japan, February 2019.
  • A Markov chain sampler for knot diagrams.
    Special Session on Mathematical Methods for the Study of the Three Dimensional Structure of Biopolymers (AMS Fall Western Sectional Meeting 2018).
    San Francisco State University, San Francisco, CA, October 2018.
  • Slipknots in unknot diagrams.
    2018 Summer Conference on Topology and its Applications.
    Western Kentucky University, Bowling Green, KY, July 2018.
  • Random Knot Diagrams: New results on open questions.
    Deguchi Laboratory Seminar.
    Ochanomizu University, Tokyo, Japan, May 2018.
  • Monte Carlo sampling of knot diagrams (workshop tutorial).
    Approximate Enumeration of Polygons, Polymers and Link Diagrams.
    University of British Columbia, Vancouver, BC, March 2018.
  • An efficient Markov chain sampler for plane curves.
    Discrete Math Seminar.
    University of British Columbia, Vancouver, BC, March 2018.
  • Slipknotting in random knot diagrams.
    The Geometry and Topology of Knotting and Entanglement in Proteins.
    Casa Matematica Oaxaca (CMO), Oaxaca, Mexico, November 2017.
    Video available here.
  • A Sumners-Whittington result for knot diagrams.
    Means, Methods, and Results in the Statistical Mechanics of Polymeric Systems II.
    Fields Institute, Toronto, ON, June 2017.
    Video available here.
  • A Markov chain sampler for knot diagrams.
    Special Session on Invariants of Knots, Links, and 3-manifolds (AMS Spring Eastern Sectional Meeting 2017).
    Hunter College, New York, NY, May 2017.
  • Slipknotting in the Knot Diagram Model.
    Special Session on Knot Theory and its Applications (AMS Spring Southeast Sectional Meeting 2017).
    College of Charleston, Charleston, SC, March 2017.
  • Slipknotting in the Knot Diagram Model.
    MAA Invited Paper Session on Random Polygons and Knots (Joint Mathematics Meetings 2017).
    Atlanta, GA, January 2017.
  • Random knots in physics and biology.
    Annual Math and Physics Lecture.
    Piedmont College, Demorest, GA, November 2016.
  • Asymptotic laws for knot diagrams.
    28th International Conference on Formal Power Series and Algebraic Combinatorics.
    Vancouver, BC, July 2016.
  • Asymptotic laws for knot diagrams.
    Graduate Student Topology and Geometry Conference 2016.
    IU Bloomington, Bloomington, IN, April 2016.
  • A robotics-based calculus class.
    MAA Session on Mathematical Modeling in the Undergraduate Classroom (Joint Mathematics Meetings 2016).
    Seattle, WA, January 2016.
  • Asymptotic laws for knot diagrams.
    AMS Session on General Topics (Joint Mathematics Meetings 2016).
    Seattle, WA, January 2016.
  • Asymptotic laws for knot diagrams.
    Geometry Seminar.
    Tulane University, New Orleans, LA, October 2015.
  • Asymptotics of random knot diagrams.
    Special Session on Algebraic and Combinatorial Structures in Knot Theory (AMS Fall Western Sectional Meetings 2015).
    CSU Fullerton, Fullerton, CA, October 2015.
  • Asymptotics of random knot diagrams.
    Special Session on Topological Combinatorics (AMS Fall Southeastern Sectional Meetings 2015).
    University of Memphis, Memphis, TN, October 2015.
  • Asymptotic laws for knot diagrams.
    Discrete Math Seminar.
    University of British Columbia, Vancouver, BC, September 2015.
  • Asymptotic laws for knot diagrams.
    Discrete Math Seminar.
    Simon Fraser University, Burnaby, BC, September 2015.
  • Knot diagrams and blossom trees.
    PIMS-USASK Graduate Summer School on Applied Combinatorics.
    University of Saskatchewan, Saskatoon, SK, May 2015.
  • Random knot diagrams.
    Special Session on Inverse Problems and Related Mathematical Methods in Physics (AMS Spring Western Sectional Meetings).
    University of Nevada, Las Vegas, NV, April 2015.
  • Packets, Solving Symmetries, and Sudoku.
    With Malcolm Rupert.
    AMS-MAA-SIAM Special Session on Research in Mathematics by Undergraduates and Students in Post-Baccalaureate Programs (Joint Mathematics Meetings 2011).
    New Orleans, LA, January 2011.
  • Packets, Solving Symmetries, and Sudoku.
    With Malcolm Rupert.
    Young Mathematicians Conference.
    The Ohio State University, Columbus, OH, August 2010.

Internal

  • Statistical mechanics of knot diagrams.
    Postdoc Seminar.
    Colorado State University, Fort Collins, CO, March 2018.
  • A Markov chain Monte Carlo sampler for knot diagrams.
    Geometry Seminar.
    University of Georgia, Athens, GA, March 2017.
  • Patterns in knot diagrams.
    Geometry Seminar.
    University of Georgia, Athens, GA, August 2016.
  • The quantum harmonic oscillator.
    Geometry Seminar.
    University of Georgia, Athens, GA, October 2015.
  • Asymptotic laws for knot diagrams.
    Geometry Seminar.
    University of Georgia, Athens, GA, September 2015.
  • How to count (a quick glance at analytic combinatorics).
    Graduate Student Seminar.
    University of Georgia, Athens, GA, September 2015.
  • Asymptotics of knot and link diagrams.
    Mock AMS Conference.
    University of Georgia, Athens, GA, July 2015.
  • Virtual Knot Theory.
    Graduate Student Seminar.
    University of Georgia, Athens, GA, February 2015.
  • Random Planar Diagrams.
    Geometry Seminar.
    University of Georgia, Athens, GA, January 2014.
  • The Poincaré homolgy sphere as the link of a singularity.
    Graduate Student Topology Seminar.
    University of Georgia, Athens, GA, November 2014.
  • Discrete Ricci Flow.
    Research Group on Minimal Surfaces.
    University of Georgia, Athens, GA, November 2014.
  • The Tropical Grassmannian.
    Mock AMS Conference.
    University of Georgia, Athens, GA, June 2014.
  • Hope for slackers: Playing games to prove theorems.
    Mock AMS Conference.
    University of Georgia, Athens, GA, June 2013.
  • The Classification of Surfaces.
    Mock AMS Conference.
    University of Georgia, Athens, GA, June 2012.
  • Vinogradov's generalization of a theorem of Aubry-Thue.
    VIGRE Research Group on Minkowski's Geometry of Numbers.
    University of Georgia, Athens, GA, March 2012.

Previous Research