Elizabeth Denne
Department of Mathematics
Washington & Lee University
202 Robinson Hall




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, my 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:
 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:
 Folded ribbon knots in the plane, with Shivani Aryal,
Eleanor Conley, Shorena Kalandarishvili, Emily Meehan and Rebecca Terry.
 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 flatribbons 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:
 Transversality theorems for configuration spaces and
the "squarepeg" problem,
with Jason Cantarella and John McCleary.
We prove that C^1smooth Jordan curves have inscribed
squares and then extend this result to curves of finite total
curvature without cusps. We also discuss curves in R^n.
 Alternating quadrisecants of knots.
I prove that every nontrivial tame knot has an
essential alternating quadrisecant. Alternating quadrisecants capture the
knottedness of a knot. Their existence implies the FaryMilnor
theorem that every knot has total curvature at least 4π.
Accepted/Published:
 Quadrisecants and essential secants of knots: with applications to the geometry
of knots. (August 2016) Accepted for publication in New directions in Geometric and Applied Knot Theory. Simon Blatt, Philipp Reiter, and Armin Schikorra, editors. De Gruyter, to appear in 2017.
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. (July 2016) Accepted for publication in
a Contemporary Mathematics volume: Knots, Graphs,
Algebra & Combinatorics, edited by E. Flapan, Allison Henrich, A. Kaestner, and S. Nelson.
We give an upper bound of ncot(π/n) for the ribbonlength of nstick unknots.
We prove that the minimum ribbonlength for a 3stick unknot with the same type of fold at
each vertex is $3\sqrt{3}$, and such a minimizer is an equilateral triangle.
 From Molecules to the Universe: an Introduction to Topology,
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.
 The distortion of a knotted curve. Joint with J.M.
Sullivan. Proc. Amer. Math. Soc. 137 no. 3 2009, pp 11391148.
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 R^3 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 163174
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 126.
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, Univeristy of Illinois at UrbanaChampaign. 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. 128138).
 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.629672.
 Of interest is the way Pannwitz proves the existence of
quadrisecants. Note that G. Kuperberg (J. Knot Theory
Vol. 3 No. 1 (1994) p. 4150) and C. Schmitz (Geom.
Dedicata 71 p. 8390, 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.)

