Summary:
I am currently looking at geometric properties of knots
and links. In particular, I am interested in quadrisecants of
knots. The existence
of essential alternating quadrisecants has implications for geometric
properties such as the total curvature of a knot. I am also
interested in optimal geometry, especially the difficult problem of
optimizing the length of thick knots.
| Mathematics Talks: |
Hampshire College Summer Studies in Mathematics. Prime time talk 7/2/09
- The talk on geometric knot theory can be found here (.pdf file 2.1Mb)
- Some advice I give to Smith students about applying to graduate school can be found here
here.
- The AMS has very good information about applying to graduate school found
here.
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| Publications: |
In preparation:
- On the ``Squarepeg'' problem. Joint
with J. Cantarella and J. McCleary. In preparation.
We prove that C^1 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.
- On flat-ribbon links in R^2. Joint with J.M. Sullivan
and N. Wrinkle.
We develop a theory of flat-ribbons in R^2. 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:
- Alternating quadrisecants of knots. See arxiv:math.GT/0510561.
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 4π.
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 in R^3 has distortion 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. 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
quadrisecants to dramatically improve the known lower bounds on
ropelength. Our theoretical results are extremely close to
computational estimates of the ropelength of small crossing knots.
Ph.D. Thesis:
- Alternating Quadrisecants of Knots.
Ph.D. Thesis, Univeristy of Illinois at Urbana-Champaign. May 2004.
Thesis in pdf
format (805Kb). (Note: 130 pages long.)
Thesis in ps
format (2Mb).
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| Research Experiences for Undergraduates: |
- Spring 2008: Supervised an undergraduate researching the minimum distance energy of knots. Smith College.
- Summer 2004 and 2005: Supervised an undergraduate first in an introductory project on knot theory and then in a more advanced project based around an open question on the supercrossing number of knots. Harvard University.
- Summer 2001 and 2002:
Associate mentor for NSF funded illiMath2001 and illiMath2002 REUs in geometry in the Mathematics
Department at University of Illinois, Urbana-Champaign.
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| Translations: |
- Here is a translation of Istvan Fary's Paper from French (On the Total Curvature of a Nonplanar Knotted Curve). It 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 to the translation or suggestions
for a better translation.
- Here is a translation of Erika Pannwitz's Paper from German (An Elementary Geometrical Property of Links and Knots). It is in pdf
format. (Last modified 5th June 2004.)
Eine elementargeometrische Eigenshaft von Verschlingungen und Knoten.
Math. Annal. 108 (1933), p.629-672.
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.)
- 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!
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