RESEARCH Summary: My Research Statement in pdf format. 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. Publications: In preparation: On the ``Squarepeg'' problem. Joint with J. Cantarella and J. McCleary. Preprint 2006. 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. Criticality for flat-ribbons. Joint with J.M. Sullivan and N. Wrinkle. We develop a theory of criticality for flat-ribbons: ribbons of fixed width about curves immersed in the plane. We examine the critical configurations of several knot and link types. Submitted: The distortion of a knotted curve. Joint with J.M. Sullivan. See arXiv:math.GT/049438v2. 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. 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: Convergence and isotopy for graphs of finite total curvature. Joint with J.M. Sullivan. To appear in Oberwolfach Lecture Notes on "Discrete Differential Geometry". 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 p.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). Research Experiences for Undergraduates: 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: Over summers 2001 & 2002 I was an associate mentor for the NSF funded VIGRE Research Experience for Undergraduates group in geometry in the Mathematics Department at University of Illinois, Urbana-Champaign. This involved helping students try to solve mathematical problems on their own, in addition to exposing them to new mathematics. In summer 2002 my students produced a wonderful java applet to explore the crossing map of knots. To see this click here. For more information on the illiMath2001 REU go to illiMath2001 and for illiMath2002 REU go to illiMath2002. 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!