On the minimum ropelength of knots and links

Inventiones Mathematicae

Volumn 150, Issue 2, 2002, Pages 257-286

  Cantarella, Jason ^a   Kusner, Robert B ^b   Sullivan, John M ^c  

^a UNIVERSITY OF GEORGIA   (United States)

^b UNIVERSITY OF MASSACHUSETTS   (United States)

^c UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035981373     PISSN: 00209910     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00222-002-0234-y     Document Type: Article

References (58)

1. [BL] Sandeep N. Bhatt, F. Thomson Leighton. A framework for solving VLSI graph layout problems. J. Comput. and Systems Sci. 28(1984), 300-343
2. [BM] Leonard M. Blumenthal, Karl Menger. Studies in Geometry. W.H. Freeman, San Francisco, 1970
3. [Buc] Greg Buck. Four-thirds power law for knots and links. Nature 392(March 1998), 238-239
4. [BS] Greg Buck, Jon Simon. Thickness and crossing number of knots. Topol. Appl. 91(1999), 245-257
5. [Cǎl1] George Cǎlugǎreanu. L'intégrale de Gauss et l'analyse des nœuds tridimensionnels. Rev. Math. Pures Appl. 4(1959), 5-20
6. [Cǎl2] George Cǎlugǎreanu. Sur les classes d'isotopie des noeuds tridimensionnels et leurs invariants. Czechoslovak Math. J. 11(1961), 588-625
7. [Cǎl3] George Cǎlugǎreanu. Sur les enlacements tridimensionnels des courbes fermées. Com. Acad. R. P. Romîne 11(1961), 829-832
8. [Can] Jason Cantarella. A general mutual helicity formula. R. Soc. Lond. Proc. Sen A Math. Phys. Eng. Sci. 456(2000), 2771-2779
9. [CKKS] Jason Cantarella, Greg Kuperberg, Robert B. Kusner, John M. Sullivan. The second hull of a knotted curve. ArXiv eprint Math. GT/0204106
10. [CKS] Jason Cantarella, Robert Kusner, John Sullivan. Tight knot values deviate from linear relation. Nature 392(1998), 237
11. [Che] Shiing Shen Chern. Curves and surfaces in euclidean space. In Shiing Shen Chern, editor, Studies in Global Geometry and Analysis, pages 16-56. Math. Assoc. Amer., 1967
12. [DS1] Isabel Darcy, De Witt Sumners. A strand passage metric for topoisomerase action. In KNOTS '96 (Tokyo), pages 267-278. World Sci. Publishing, River Edge, NJ, 1997
13. [DS2] Isabel Darcy, De Witt Sumners. Applications of topology to DNA. In Knot theory (Warsaw, 1995), pages 65-75. Polish Acad. Sci., Warsaw, 1998
14. [dFPP] Hubert de Fraysseix, János Pach, Richard Pollack. How to draw a planar graph on a grid. Combinatorica 10(1990), 41-51
15. [Dia] Yuanan Diao. The lower bounds of the lengths of thick knots. J. Knot Theory Ramif. To appear
16. [DEJvR] Yuanan Diao, Claus Ernst, E. J. Janse van Rensburg. Upper bounds on linking number of thick links. J. Knot Theory Ramifications 11(March 2002), 199-210
17. [ES] Michael Edelstein, Binyamin Schwarz. On the length of linked curves. Israel J. Math. 23(1976), 94-95
18. [EWW] Tobias Ekholm, Brian White, Daniel Wienholtz. Embeddedness of minimal surfaces with total boundary curvature at most 4π. Ann. of Math. 155(2002), 109-234
19. [Fár] István Fáry. Sur la courbure totale d'une courbe gauche faisant un noeud. Bull. Soc. Math. France 77(1949), 128-138
20. [Fed] Herbert Federer. Curvature measures. Trans. Amer. Math. Soc. 93(1959), 418-491
21. [FH] Michael H. Freedman, Zheng-Xu He. Divergence free fields: energy and asymptotic crossing number. Ann. of Math. 134(1991), 189-229
22. [Ful] F. Brock Fuller. Decomposition of the linking number of a closed ribbon: a problem from molecular biology. Proc. Nat. Acad. Sci. (USA) 75(1978), 3557-3561
23. [Gag] Michael E. Gage. A proof of Gehring's linked spheres conjecture. Duke Math. J. 47(1980), 615-620
24. [GH] Mohammad Ghomi, Ralph Howard. Unfoldings of space curves. In preparation
25. [GdlL] Oscar Gonzalez, Rafael de la Llave. Existence of ideal knots. J. Knot Theory Ramif. To appear
26. [GM] Oscar Gonzalez, John H. Maddocks. Global curvature, thickness, and the ideal shapes of knots, Proc. Nat. Acad. Sci. (USA) 96(1999), 4769-4773
27. [GM+] Oscar Gonzalez, John H. Maddocks, Friedmann Schuricht, Heiko von der Mosel. Global curvature and self-contact of nonlinearly elastic curves and rods. Calc. Var. Partial Differential Equations 14(2002), 29-68
28. [Gro] Mikhael Gromov. Filling Riemannian manifolds. J. Diff. Geom. 18(1983), 1-147
29. [Har] Shelly L. Harvey. Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm. Preprint, 2002
30. [He] Zheng-Xu He. On the crossing number of high degree satellites of hyperbolic knots. Math. Res. Lett. 5(1998), 235-245
31. [Joh] Heather Johnston. An upper bound on the minimal edge number of an n-crossing lattice knot. Preprint
32. +] Vsevolod Katritch, Jan Bednar, Didier Michoud, Robert G. Scharein, Jacques Dubochet, Andrzej Stasiak. Geometry and physics of knots. Nature 384(November 1996), 142-145
33. +] Vsovolod Katritch, Wilma K. Olson, Piotr Pieranski, Jacques Dubochet, and Andrzej Stasiak. Properties of ideal composite knots. Nature 388(July 1997), 148-151
34. [KS] Robert B. Kusner, John M. Sullivan. On distortion and thickness of knots. In S. Whittington, D. Sumners, T. Lodge, editors, Topology and Geometry in Polymer Science, IMA Volumes in Mathematics and its Applications, 103, pages 67-78. Springer, 1997. Proceedings of the IMA workshop, June 1996
35. [Lan] Ben Laurie. Annealing ideal knots and links: Methods and pitfalls. In A. Stasiak, V. Katritch, L. Kauffman, editors, Ideal Knots, pages 42-51. World Scientific, 1998
36. +] Ben Laurie, Vsevolod Katritch, Jose Sogo, Theo Koller, Jacques Dubochet, and Andrzej Stasiak. Geometry and physics of catenanes applied to the study of DNA replication. Biophysical Journal 74(1998), 2815-2822
37. [Lei1] F. Thomson Leighton. Complexity Issues in VLSI. MIT Press, Cambridge, MA, 1983
38. [Lei2] F. Thomson Leighton. New lower bound techniques for VLSI. Math. Syst. Theory 17(1984), 47-70
39. [LSDR] Richard A. Litherland, Jon Simon, Oguz Dummeric, Eric Rawdon. Thickness of knots. Topol. Appl. 91(1999), 233-244
40. [MCM] Curtis T. MCMullen. The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology. Ann. Scient. Éc. Norm. Sup. 35(2002). To appear
41. [MB] David Miller, Craig Benham. Fixed-writhe isotopies and the topological conservation taw for closed, circular DNA. J. Knot Theory Ramifications 5(1996), 859-866
42. [Mil] John W. Milnor. On the total curvature of knots. Ann. of Math. 52(1950), 248-257
43. [Oss] Robert Osserman. Some remarks on the isoperimetric inequality and a problem of Gehring. J. Analyse Math. 30(1976), 404-410
44. [Pie] Piotr Pieranski. In search of ideal knots. In A. Stasiak, V. Katritch, L. Kauffman, editors, Ideal Knots, pages 20-41. World Scientific, 1998
45. [Raw] Eric Rawdon. Approximating the thickness of a knot. In A. Stasiak, V. Katritch, L. Kauffman, editors, Ideal Knots, pages 143-150. World Scientific, 1998
46. [Res1] Yuriǐ G. Reshemyak. On the theory of spaces with curvature no greater than K. Mat. Sb. (N.S.) 52 (94)(1960), 789-798
47. [Res2] Yuriǐ G. Reshetnyak. Inextensible mappings in a space of curvature no greater than K. Siberian Math. J. 9(1968), 683-689
48. [Sal] G. Thomas Sallee. Stretching chords of space curves. Geometriae Dedicata 2(1973), 311-315
49. [Sch1] Walter Schnyder. Embedding planar graphs on the grid. In Proc. 1st ACM-SIAM Sympos. Discrete Algorithms, pages 138-148, 1990
50. [Sch2] Achill Schürmann. On extremal finite packings. Discrete Comput. Geom. (2002). To appear
51. [SDKP] Andrzej Stasiak, Jacques Dubochet, Vsevolod Katritch, Piotr Pieranski. Ideal knots and their relation to the physics of real knots. In A. Stasiak, V. Katritch, L. Kauffman, editors, Ideal Knots, pages 20-41. World Scientific, 1998
52. +] Andrzej Stasiak, Vsevolod Katritch, Jan Bednar, Didier Michoud, Jacques Dubochet. Electrophoretic mobility of DNA knots. Nature 384(November 1996), 122
53. [SKK] Andrzej Stasiak, Vsovolod Katritch, Lou Kauffman, editors. Ideal Knots. World Scientific, 1998
54. [Sum] De Witt Sumners. Lifting the curtain: using topology to probe the hidden action of enzymes. Match (1996), 51-76
55. [Thi] Morwen B. Thistlethwaite. On the Kauffman polynomial of an adequate link. Invent. Math. 93(1988), 285-296
56. [TL] Morwen B. Thistlethwaite, W. B. Raymond Lickorish. Some links with non-trivial polynomials and their crossing numbers. Comm. Math. Helv. 63(1988), 527-539
57. [Whi1] Brian White. Half of Enneper's surface minimizes area. In Jürgen Jost, editor, Geometric Analysis and the Calculus of Variations, pages 361-367. Internat. Press, 1996
58. [Whi2] James White. Self-linking and the Gauss integral in higher dimensions. Amer. J. Math. 91(1969), 693-728