Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T10:51:23.939Z Has data issue: false hasContentIssue false
  • English
  • Français

The growth of the number of prime knots

Published online by Cambridge University Press:  24 October 2008

C. Ernst  and
D. W. Sumners
C. Ernst
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, FL 32306, U.S.A.
D. W. Sumners
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, FL 32306, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

A fundamental and interesting question in knot theory is:

Question 1. How many prime knots of n crossings are there ?

Over time, knot theorists have answered this question for n ≤ 13 by the method of exhaustion: one writes down a list of all possible knots of n crossings, and then works hard to eliminate duplications from the list [12]. A perhaps easier question is the following:

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 2 , September 1987 , pp. 303 - 315
Copyright
Copyright © Cambridge Philosophical Society 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bonahon, F.. Involutions et Fibre de Seifert dans les Variétés de dimension 3. Thèse de 3e cycle, Orsay (1979).Google Scholar
[2]Bonahon, F. and Siebenmann, L. C.. New geometric splittings of classical knots (algebraic knots). (To appear.)Google Scholar
[3]Burde, G. and Zieschang, H.. Knots (de Gruyter, 1985).Google Scholar
[4]Conway, J.. An enumeration of knots and links and some of their related properties. Computational problems in Abstract Algebra. Proc. Conf. Oxford 1967 (Pergamon Press, 1967), 329–358.Google Scholar
[5]Kauffman, L. H.. State models and the Jones polynomial. Topology (to appear).Google Scholar
[6]Michels, J. P. J. and Wiegel, F. W.. Probability of knots in a polymer ring. Phys. Lett. A 90 (1982).CrossRefGoogle Scholar
[7]Murasugi, K.. Jones polynomials and classical conjectures in knot theory. Topology (to appear).Google Scholar
[8]Schubert, H.. Knoten mit zwei brucken. Math. Z. 65 (1956), 133–170.CrossRefGoogle Scholar
[9]Siebenmann, L.. Excercices sur les noeuds rationnels. Lecture Notes Orsay (1975).Google Scholar
[10]Spengler, S. J., Stasiak, A. and Cozzarelli, N. R.. The stereostructure of knots and catenanes produced by phage λ integrative recombination: implications for mechanism and DNA structure. Cell 42 (1985), 325–334.CrossRefGoogle ScholarPubMed
[11]Sumners, D. W.. The role of knot theory in DNA research. Geometry and Topology (ed. McCrory, C. and Schifrin, T.) (Marcel Dekker, 1987), pp. 297–318.Google Scholar
[12]Thistlethwaite, M. B.. Knot tabulations and related topics. Aspects of Topology in Memory of Hugh Dowker, 1912–1982, L.M.S. Lecture Notes no. 93 (Cambridge University Press, 1985), 1–76.Google Scholar
[13]Thistlethwaite, M. B.. A spanning tree expansion of the Jones polynomial. Topology (to appear).Google Scholar
[14]Zieschang, H.. Classification of Montesinos knots. Lecture Notes in Mathematics, no. 1060 (Springer, 1984), 379–389.Google Scholar