Papers

Papers

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Projects with their own pages

Square Peg problem : Work in progress on the problem-- does every simple closed curve in the plane contain four points which form a perfect square?

Carpenter's Rule problem: A fast algorithm for unfolding linkages composed of straight line segments of fixed length in the plane without introducing self intersections.

Nth Hull problem: Computing stronger versions of the convex hull for space curves. While every plane through a point in the convex hull of a curve intersects the curve twice, every plane through a point in the n-th convex hull of a curve intersects the curve 2n times. What is the relationship between the topology of a curve and its' n-th hulls?

Preprints and Reprints

To obtain a copy of any of the papers listed below, click on the title of the paper. The preprints should be cited by arXiv number (as they are all also available from the arXiv ).

The MR number beside each (published) title gives the math review for that paper. The BibTeX citation information for all the papers is collected in cantarella.bib. Published papers use their MR numbers as BibTeX tags. Alternate BibTeX tags for the preprints appear below the entries for those papers.

  1. Knot Tightening by Constrained Gradient Descent
    Ted Ashton, Jason Cantarella, Michael Piatek, and Eric Rawdon.
    (2 MB PDF)
    Experimental Mathematics (to appear)
    arXiv:1002.1723
    acprtightening
    See also Self contact sets for 50 Tightly Knotted and Linked Tubes
    Data (designed to work with our Octrope and plCurve libraries):
    1. Atlas of Tight Knots and Links (44 MB PDF)
    2. Tight Knot and Link Vertex Coordinates (6MB VECT files)
    3. Ropelengths of Knots with <= 10 and Links with <= 9 crossings (17 KB text file)
  2. A new cohomological formula for helicity in $\R^{2k+1}$ reveals the effect of a diffeomorphism on helicity
    Jason Cantarella and Jason Parsley.
    Journal of Geometry and Physics 60 (2010), p. 1127-1155.
    arxiv: math.GT/09031465
    (519 K PDF)
    cphelicity
  3. Visualizing the tightening of knots
    Jason Cantarella, Michael Piatek, and Eric Rawdon.
    Proceedings of IEEE Visualization 2005, p. 575-582.
    (3417 K PDF)
    cprvis
  4. A fast octree-based algorithm for computing ropelength
    Ted Ashton and Jason Cantarella.
    In Physical and Numerical Models in Knot Theory
    and their Application to the Life Sciences    , World Scientific Press (2005), p. 323-341
    (169 K PDF)
    arXiv: math.DG/0409416
    See also the Octrope homepage.
    MR2197947
  5. TSNNLS: A solver for large sparse least squares problems with non-negative variables
    Jason Cantarella and Michael Piatek.
    Preprint. (2004).
    (92 K PDF)
    arXiv: cs.MS/0408029
    See also the Tsnnls homepage.
    tsnnls
  6. Criticality for the Gehring Link Problem
    Jason Cantarella, Joseph H.G. Fu, Rob Kusner, John M. Sullivan, and Nancy Wrinkle.
    Geometry and Topology 10 (2006), p. 2055-2116.
    arXiv: math.DG/0402212
    (888 K PDF)
    MR2284052
  7. Upper Bounds for Ropelength as a function of Crossing Number
    Jason Cantarella, X.W. Faber, and Chad A. Mullikin.
    Topology and its Applications 135 (2003), no. 1-3, p. 253-264.
    (arXiv: math.GT/0210245)
    (1.6 M PDF)
    ScienceDirect version of this paper.
    MR2024960
  8. An Energy-Driven Approach to Linkage Unfolding
    Jason Cantarella, Erik Demaine, Hayley Iben and James O'Brien.
    SCG '04: Proceedings of the twentieth annual symposium on Computational geometry, p. 134-143
    (212 K PDF)

    Abstract (not posted) in Proceedings of the 12th Annual DIMACS Fall Workshop
    on Computational Geometry, Piscataway, New Jersey, November 14-15, 2002.

    Movies and Java Applet for this paper
    cdob
  9. The Second Hull of a Knotted Curve
    Jason Cantarella, Greg Kuperberg, Robert B. Kusner, and John M. Sullivan.
    American Journal of Mathematics 125 (2003) no. 6, p. 1335-1348.
    (arXiv: math.GT/0204106)
    (818 K PDF)
    MR2018663
  10. On Comparing the Writhe of a Smooth Curve to the Writhe of an Inscribed Polygon
    Jason Cantarella.
    SIAM Journal of Numerical Analysis 42 (2005) no. 4, p. 1846-1861.
    (arXiv: math.DG/0202236)
    (601 K PS) (151 K PDF)
    MR2139226
  11. Vector Calculus and the Topology of Domains in 3-Space
    Jason Cantarella, Dennis DeTurck, and Herman Gluck.
    American Mathematical Monthly 109 (2002) no. 5.
    (2,580 KB PDF)
    MR2003c:53023
  12. On the Minimum Ropelength of Knots and Links
    Jason Cantarella, Robert B. Kusner, and John M. Sullivan.
    Inventiones Mathematicae 150 (2002) no. 2, p. 257-286.
    Springer LINK Version (411 K PDF)
    MR2003h:58014
  13. Circles Minimize Most Knot Energies
    Aaron Abrams, Jason Cantarella, Joe Fu, Mohammad Ghomi, and Ralph Howard.
    Topology 42 (2002) no. 2, p. 381-394.
    (368 K PS)
    MR2004f:58014
    This paper was one of the 5 most downloaded articles in the
    journal Topology during January-August of 2004.
  14. The Biot-Savart operator for application to knot theory, fluid dynamics, and plasma physics
    Jason Cantarella, Dennis DeTurck, and Herman Gluck.
    Journal of Mathematical Physics 42 (2001), no. 2, p. 876-905.
    (293 K PDF)
    MR2002e:78002
  15. Isoperimetric problems for the helicity of vector fields and the Biot-Savart and curl operators
    Jason Cantarella, Dennis DeTurck, Herman Gluck, and Mikhail Teytel.
    Journal of Mathematical Physics 41 (2000), no. 8, p. 5615-5641.
    (888 K PDF)
    MR2001f:78010
  16. A General Cross-Helicity Formula
    Jason Cantarella.
    Proceedings of the Royal Society, Series A 456 (2000) no. 2003, p. 2771-2779
    (401 K PS)
    MR2002f:53002
  17. Upper Bounds for the Writhing of Knots and the Helicity of Vector Fields
    Jason Cantarella, Dennis DeTurck, and Herman Gluck.
    Proceedings of the Conference in Honor of the 70th Birthday of Joan Birman
    Jane Gilman, Xiao-Song Lin, William Menasco (eds)
    International Press, AMS/IP Series on Advanced Mathematics (2000)
    (1,708 K PS) (314 K PDF)
    MR2003j:58018
  18. Tight Knot Values Deviate From Linear Relation
    Jason Cantarella, Robert B. Kusner, and John M. Sullivan.
    Nature 392, March 19, 1998, p. 237.
    (720 K compressed PS)
    cksnature
  19. Nontrivial Embeddings of Polygonal Intervals and Unknots in 3-Space
    Jason Cantarella and Heather Johnston.
    J. Knot Theory Ramifications, Vol. 7, No. 8 (1998) p. 1027-1039.
    (195 K PDF) (370 K PS)
    MR99m:57002
  20. Eigenvalues and Eigenfields of the Biot-Savart and Curl Operators
    on Spherically Symmetric Domains
    Jason Cantarella, Dennis DeTurck, Herman Gluck, and Misha Teytel.
    Physics of Plasmas 7(7), 2000. pp.2766-2775.
    (1,586 K PDF)
    MR2001b:76094
  21. Influence of Geometry and Topology on Helicity
    Jason Cantarella, Dennis DeTurck, Herman Gluck, and Misha Teytel.
    In Magnetic Helicity in Space and Laboratory Plasmas,
    Michael Brown, Richard Canfield and Alexei Pevtsov (eds),
    Geophysical Monographs 111, American Geophysical Union. (1999)
    (164 K PDF) (879 K PS)
    influence
  22. Topological Structure of Stable Plasma Flows
    Jason Cantarella.
    Ph.D. Thesis, University of Pennsylvania, 1999.
    (357 K PDF)
    cantarellathesis
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