## 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:

- 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

## Publications:

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

**In Preparation:**

with Shivani Aryal, Eleanor Conley, Corinne Joireman, Shorena Kalandarishvili, Emily Meehan and Allison Young.*Folded ribbon knots in the plane,*with J.M. Sullivan and N. Wrinkle.*Medial axis for immersed disks,*with J.M. Sullivan and N. Wrinkle.*Ribbonlength for knot diagrams,*

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.*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.

**Submitted:**

**Folded ribbon knots in the plane.**

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.with Jason Cantarella and John McCleary.*Transversality theorems for configuration spaces and the “square-peg” problem,*

We prove that \(C^1\)-smooth Jordan curves have inscribed squares and then extend this result to curves of finite total curvature without cusps. We also discuss curves in \(\mathbb{R}^n\).*Alternating quadrisecants of knots.*

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:**

. (August 2016) Accepted for publication in*Quadrisecants and essential secants of knots: with applications to the geometry of knots**New directions in Geometric and Applied Knot Theory.*Simon Blatt, Philipp Reiter, and Armin Schikorra, editors. De Gruyter, to appear in 2018.

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.with Mary Kamp, Rebecca Terry, and Xichen (Catherine) Zhu. In*Ribbonlength of folded ribbon unknots in the plane,**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.with Erica Flapan & 17 other members of the Undergraduate Faculty Program at PCMI (July 2011). This is an introductory undergraduate textbook on topology. Published by the American Mathematical Society, 2016.*From Molecules to the Universe: an Introduction to Topology,*Joint with J.M. Sullivan. Proc. Amer. Math. Soc.*The distortion of a knotted curve.***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.Joint with J.M. Sullivan. In “Discrete Differential Geometry” Birkhouser 2008 pp 163-174*Convergence and isotopy for graphs of finite total curvature.*

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.Joint with Y. Diao, J.M. Sullivan. Geometry and Topology vol. 10, 2006 pp 1-26.*Quadrisecants give new bounds for ropelength.*

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.)