E. Denne - Research

Research

I am interested in Geometric Knot Theory. My research uses topological knot invariants to answer questions about geometric properties of knots and links. I use a mix of geometry, topology, analysis, and differential topology in my research. I am interested in topological questions about quadrisecants of knots. The existence of essential alternating quadrisecants has implications for geometric properties such as the total curvature, ropelength and distortion of a knot. A key tool in this research is configuration spaces. I am also interested in optimal geometry, especially the difficult problem of optimizing the length of thick knots. My research has many applications to the natural sciences - biology, physics and engineering.

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

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

Thanks to my wonderful collaborators:

Accepted/Published:

The distortion of a knotted curve. Joint with J.M. Sullivan. Proc. Amer. Math. Soc. 137 no. 3 2009, pp 1139--1148. See arXiv:math.GT/049438.
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 has distortion at least 5pi/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. See arXiv:math.GT/0606008
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 essential quadrisecants to greatly improve the known lower bounds on ropelength. Our theoretical results are within 5% of the numerical estimate for the ropelength of the presumed tight (ropelength minimized) trefoil knot.

Submitted:

Alternating quadrisecants of knots. See arxiv:math.GT/0510561.
I prove that every non-trivial tame knot has an alternating quadrisecant, and every non-trivial knot of finite total curvature 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 4π.

In Preparation:

Transversality theorems for configuration spaces and applications to the ``square peg'' problem. Joint with J. Cantarella and J. McCleary. In preparation.
We solve a number of problems using the language of configuration spaces. To do this we prove (and use) transversality theorems for configuration spaces. For example, we show that any generic smooth space curve has an odd number of square-like quadrilaterals (with equal sides and equal diagonals) inscribed in it. Thus, when k = 2, this implies that any smooth simple closed curve has at least one inscribed square: a “square peg” in a round hole.
Animations from this project may be found here .
Ribbonlength for knot diagrams. Joint with J.M. Sullivan and N. Wrinkle. In preparation.
We develop a theory of flat-ribbons in the plane. These are ribbons of fixed width about a knot diagram, i.e. about curves immersed in the plane. This is the two-dimensional analogue of the ropelength problem. We show the ribbon width imposes structure on the knot diagram; for example giving bounds on the local curvature, and information about the structure of the medial axis in regions bounded by the diagram. 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.
From Molecules to the Universe: an Introduction to Topology. Joint with Erica Flapan the 17 other members of the Undergraduate Faculty Program at PCMI, July 2011. In preparation. An introductory undergraduate textbook on topology.

Alternating Quadrisecants of Knots.
Ph.D. Thesis, Univeristy of Illinois at Urbana-Champaign. May 2004.

Thesis in pdf format (805Kb). (Note: 130 pages long.)

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 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.)