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Research

Summary

I am interested in Geometric Knot Theory. My research uses topological knot invariants to answer questions about the geometry of knots. My research has many applications to the natural sciences – biology, physics and engineering. I use a mixture of geometry, topology and analysis in my research.

Some of my projects have been with undergraduates. Click here for more information.

This page contains a list of my publications and preprints, PhD Thesis, and the English translations of two papers.

  Knots from Knotplot

Collaborators

Thanks to my wonderful collaborators:

Thanks also to my fabulous student collaborators:

  • John Carr Haden, Troy Larsen
  • Corinne Joireman and Allison Young
  • Emily Jaekle and Ryan McDonnell
  • Mary Kamp and Xichen (Catherine) Zhu
  • Eleanor Conley, Emily Meehan and Rebecca Terry
  • Shivani Aryal, Shorena Kalandarishvili

Recent Talks

Publications

Copies of all my papers may be found on the math arXiv.

In Preparation:

  • Quadrisecants and unknotting number of knots.
    I show that any generic nontrivial polygonal knot K has at least u(K) alternating knots, where u(K) is the unknotting number of K.
  • Medial axis for immersed disks, with J.M. Sullivan and N. Wrinkle.
  • Ribbonlength for knot diagrams, with J.M. Sullivan and N. Wrinkle.
    We develop a theory of flat-ribbons in the plane. These are ribbons of fixed width about curves immersed in the plane. We also provide examples of critical configurations of several knot and link types.

Submitted:

  • Square-like quadrilaterals inscribed in embedded space curves, with Jason Cantarella and John McCleary. arXiv:2103.13848 (math.DG) 11 pages, 2 figures.  We prove that all curves of finite total curvature without cusps have an inscribed square-like quadrilateral (one with equal sides and equal diagonals).
  • Alternating quadrisecants of knots. arXiv:0510561 (math.GT) 37 pages, 22 figures.
    I prove that every non-trivial tame knot has an essential alternating quadrisecant. Alternating quadrisecants capture the knottedness of a knot. Their existence implies the Fary-Milnor theorem that every knot has total curvature at least \( ( 4 \pi )  \).

Accepted

Published

  • Linking number and folded ribbon unknots, with Troy Larsen. Published by the Journal of Knot Theory and its Ramifications Vol. 32, Issue No. 01, Article No. 2350003, Year 2023 (42 pages). https:///dx.doi.org/10.1142/S0218216523500037 We study Kauffman’s model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a folded ribbon knot. The folded ribbon knot is also a framed knot, and the ribbon linking number is the linking number of the knot and one boundary component of the ribbon. We find the minimum folded ribbonlength for 3-stick unknots with ribbon linking numbers \( \pm 1\) and \(\pm 3\), and we prove that the minimum folded ribbonlength for n-gons with obtuse interior angles is achieved when the n-gon is regular. Among other results, we prove that the minimum folded ribbonlength of any folded ribbon unknot which is a topological annulus with ribbon linking number \(\pm n\) is bounded from above by \( 2n \).
  • Ribbonlength of families of folded ribbon knots, with John Carr Haden, Troy Larsen, and Emily Meehan. Published in Involve, a journal of mathematics 15-4 (2022), 591–628. DOI 10.2140/involve.2022.15.591. We give upper bounds on the folded ribbonlength of 2-bridge, (2,p) torus, twist, and pretzel knots, and these upper bounds turn out to be linear in crossing number. We give a new way to fold (p,q) torus knots, and show that their folded ribbonlength is bounded above by 2p. This means, for example, that the trefoil knot can be constructed with a folded ribbonlength of 6. We then show that any (p,q) torus knot K has a constant \(c>0\), such that the folded ribbonlength is bounded above by \(c\cdot Cr(K)^{1/2}\), providing an example of an upper bound on folded ribbonlength that is sub-linear in crossing number.
  • Families of similar simplices inscribed in most smoothly embedded spheres, with Jason Cantarella and John McCleary.  Published in Forum of Mathematics, Sigma, Volume 10, 2022, e101. DOI: https://doi.org/10.1017/fms.2022.88  Let \( \Delta \) denote a non-degenerate k-simplex in \( R^k\).  The set Sim(\(\Delta\)) of simplices in \( R^k\) similar to \( \Delta \) is diffeomorphic to \( O(k)\times [0,\infty) \times R^k\), where the factor in \( O(k) \) is a matrix called the pose.  Among (k−1)-spheres smoothly embedded in \( R^k\) and isotopic to the identity, there is a dense family of spheres, for which the subset of Sim(\(\Delta\)) of simplices inscribed in each embedded sphere contains a similar simplex of every pose. Further, the intersection of Sim(Delta) with the configuration space of k+1 distinct points on an embedded sphere is a manifold whose top homology class maps to the top class in \( O(k) \) via the pose map.
  • Configuration Spaces, Multijet Transversality, and the Square-peg Problem, with Jason Cantarella and John McCleary. Published in the Illinois Journal of Mathematics vol. 66, no. 3 pages 385-420, September 2022. https://doi.org/10.1215/00192082-10120454
    We prove a transversality “lifting property” for compactified configuration spaces as an application of the multijet transversality theorem. We use this setup to provide an attractive proof of the square-peg problem: there is a dense family of smoothly embedded circles in the plane where each simple closed curve has an odd number of inscribed squares, and there is a dense family of smoothly embedded circles in \( R^n\) where each simple closed curve has an odd number of inscribed square-like quadrilaterals.
  • Ribbonlength and crossing number of folded ribbon knots. Published in Journal of Knot Theory and Its Ramification Vol 30, Issue 4, Article 2150028, 2021. DOI: 10.1142/S0218216521500280. I show for any knot or link type that there exist constants \(c_1, c_2>0\) such that the ribbonlength is bounded above by \(c_1\cdot Cr(K)^2\), and also by \(c_2\cdot Cr(K)^{3/2}\). I use a different method for each bound. The constant \(c_1\) is quite small in comparison to \(c_2\), and the first bound is lower than the second for knots and links with \(Cr(K)\leq 12,748\).
  • The Mathematics of Tie Knots, with Corinne Joireman and Allison Young.  Published in Involve – a journal of mathematics Vol. 14 (2021), No. 2, 241–270. DOI 10.2140/involve.2021.14.241 We classify the knot type of all 85 tie knots given by T. Fink and Y. Mao. We describe how the unknot, left and right trefoil, twist knots and (2,p) torus knots can be recognized from their sequence of moves. We also view tie knots as a family within the set of all knots. Among other results, we prove that any tie knot is prime and alternating.
  • Folded ribbon knots in the plane. Published in the Encyclopedia of Knot Theory (chapter 88) December 29, 2020.  Editors: Colin Adams, Erica Flapan, Allison Henrich, Louis Kauffman, Lew Ludwig, and Sam Nelson, editors. Publisher is Taylor & Francis (2020).
    This survey article reviews Kauffman’s model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a ribbon, and the ribbonlength problem asks to minimize the ribbonlength for a given knot type. We give a summary of known results.
  • Quadrisecants and essential secants of knots: with applications to the geometry of knots.  In New directions in Geometric and Applied Knot Theory. Simon Blatt, Philipp Reiter, and Armin Schikorra, editors, pp 138-158 De Gruyter (2018). https://doi.org/10.1515/9783110571493
    A quadrisecant line is one which intersects a curve in at least four points, while an essential secant captures something about the knottedness of a knot. This survey article gives a brief history of these ideas, and shows how they may be applied to questions about the geometry of a knot via the total curvature, ropelength and distortion of a knot.
  • Ribbonlength of folded ribbon unknots in the plane, with Mary Kamp, Rebecca Terry, and Xichen (Catherine) Zhu. In Knots, Links, Spatial Graphs and Algebraic Invariants, edited by E. Flapan, Allison Henrich, A. Kaestner, and S. Nelson. Contemporary Mathematics Vol. 689, 2017. American Mathematica Society, Providence RI, pp 37 – 51.
    We give an upper bound of \(n\cot(\pi/n)\) for the ribbonlength of n-stick unknots. We prove that the minimum ribbonlength for a 3-stick unknot with the same type of fold at each vertex is \(3\sqrt{3}\), and such a minimizer is an equilateral triangle.
  • Knots, Molecules, and the Universe: an Introduction to Topology, with Erica Flapan & 17 other members of the Undergraduate Faculty Program at PCMI (July 2011). Published by the American Mathematical Society, 2016. This is an introductory undergraduate textbook on topology. 
  • The distortion of a knotted curve. Joint with J.M. Sullivan. Proc. Amer. Math. Soc. 137 no. 3 2009, pp 1139–1148.
    Gromov defined distortion as the maximum ratio of arclength to chordlength. We use the existence of an essential secant to show that any nontrivial tame knot in \(\mathbb{R}^3\) has distortion of at least \(5\pi/3\). Examples show that distortion under 7.1 suffices to build a trefoil knot.
  • Convergence and isotopy for graphs of finite total curvature. Joint with J.M. Sullivan. In “Discrete Differential Geometry” Birkhouser 2008 pp 163-174
    Generalizing Milnor’s result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy. We also show how to obtain sharper results when the starting curve is smooth.
  • Quadrisecants give new bounds for ropelength. Joint with Y. Diao, J.M. Sullivan. Geometry and Topology vol. 10, 2006 pp 1-26.
    We use quadrisecants to greatly improve the known lower bounds on ropelength. Our theoretical results are extremely close to computational estimates of the ropelength of small crossing knots.

PhD Thesis

  • Alternating Quadrisecants of Knots.
    Ph.D. Thesis, University of Illinois at Urbana-Champaign. May 2004.
    Thesis in pdf format (805Kb). (Note: 130 pages long.) Thesis in ps format (2Mb).

Translations

On the Total Curvature of a Nonplanar Knotted Curve by Istvan Fary. The translation from French is in pdf format. (Last modified October 2001.)

  • Sur La Courbure Totale D’une Courbe Gauche Faisant un Noeud. Bull. Soc. Math. France. Vol 77, 1949 (p. 128-138).
  • Please note that I have just translated the text. There are some pictures in the paper after equation (20) – see the original paper.
  • Please email me any corrections or suggestions to improve the translation.

An Elementary Geometrical Property of Links and Knots by Erika Pannwitz. The translation from the German (with Thomas Kuhnt) is in pdf format. (Last modified 5th June 2004.)

  • Eine elementargeometrische Eigenshaft von Verschlingungen und Knoten. Math. Annal. 108 (1933), p.629-672.
  • Of interest is the way Pannwitz proves the existence of quadrisecants. Note that G. Kuperberg (J. Knot Theory Vol. 3 No. 1 (1994) p. 41-50) and C. Schmitz (Geom. Dedicata 71 p. 83-90, 1998) both repeat arguments from her paper. In particular, those arguments dealing with quadrisecants arising from trisecants with common first and third points (Kuperberg) and common first and second points (Schmitz).
  • The paper is long, so I have included the original page numbers in the margins – this should aid those who wish to consult the original paper.
  • I have just translated the text. There are some pictures in the paper not in this pdf document – see the original paper:
    Fig. 1 on p. 639 consists of the usual Reidemeister moves,
    Fig. 2 on p. 644 consists of the trefoil knot linked with an unknot. The unknot is placed about a crossing on the trefoil. It crosses over two strands, then under two strands.
    Fig. 3 on p. 644 consists of a trefoil knot together with a curve parallel to it.
    Fig. 4. on p. 645 consists of the Whitehead (or Antoine) Link.
  • This translation was done quickly. Some sentences have paraphrased the original, others have a distinct Germanic flavor to them. Please email corrections or suggestions for a smoother translation!
    • Thanks to Gyo Taek Jin for corrections!
    • Thanks for Lee Rudolph for reminding us all that Math. Annalen is now online, freely accessible. (I’m still trying to find a link to this paper that works reliably.)