The Mathematical Intelligencer

, Volume 20, Issue 4, pp 33–48

The first 1,701,936 knots

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References

  1. [AB]
    J.W. Alexander and G.B. Briggs, On types of knotted curves,Ann. Math. 28 (1927), 562–586.CrossRefMATHMathSciNetGoogle Scholar
  2. [Am]
    B. Arnold, M. Au, C. Candy, K. Erdener, J. Fan, R. Flynn, J. Hoste, R.J. Muir, and D. Wu, Tabulating alternating knots through 14 crossings,J. Knot Theory Ramifications 3(4) (1994), 433–437.CrossRefMATHMathSciNetGoogle Scholar
  3. [BS]
    F. Bonahon and L. Siebenmann, The classification of algebraic links, unpublished manuscript.Google Scholar
  4. [BZ]
    G. Burde and H. Zieschang, Knots, New York: de Gruyter (1985).MATHGoogle Scholar
  5. [Cau]
    A. Caudron, Classification des noeuds et des enlacements, Prepublication Math. d’Orsay, Orsay, France: Université Paris- Sud (1980).Google Scholar
  6. [Con]
    J.H. Conway, An enumeration of knots and links,Computational Problems in Abstract Algebra (Leech, ed.), New York: Pergamon Press (1970), 329–358.Google Scholar
  7. [Deh]
    M. Dehn, Die beiden Kleeblattschlingen,Math. Ann. 75 (1914), 402–413.CrossRefMathSciNetGoogle Scholar
  8. [DH]
    H. Doll and J. Hoste, A tabulation of oriented links,Math. Computat. 57(196) (1991), 747–761.CrossRefMATHMathSciNetGoogle Scholar
  9. [DT]
    C.H. Dowker and M.B. Thistlethwaite, Classification of knot projections,Topol. Appl. 16 (1983), 19–31.CrossRefMATHMathSciNetGoogle Scholar
  10. [EP]
    D.B.A. Epstein and R.C. Penner, Euclidean decompositions of noncompact hyperbolic manifolds,J. Diff. Geom. 27 (1988), 67–80.MATHMathSciNetGoogle Scholar
  11. [ES]
    C. Ernst and D.W. Sumners, The growth of the number of prime knots,Math. Proc. Camb. Phil. Soc. 102 (1987), 303–315.CrossRefMATHMathSciNetGoogle Scholar
  12. [GL]
    C. Gordon and J. Luecke, Knots are determined by their complements,J. Am. Math. Soc. 2 (1989), 371–415.CrossRefMATHMathSciNetGoogle Scholar
  13. [Hak]
    W. Haken, Theorie der Normalflächen,Acta Math. 105 (1961), 245–375.CrossRefMATHMathSciNetGoogle Scholar
  14. [Has1]
    M.G. Haseman, On knots, with a census of the amphicheirals with twelve crossings,Trans. Roy. Soc. Edinburgh 52 (1917), 235–255.CrossRefGoogle Scholar
  15. [Has2]
    M.G. Haseman, Amphicheiral knots,Trans. Roy. Soc. Edinburgh 52 (1918), 597–602.CrossRefGoogle Scholar
  16. [Hem]
    G. Hemion, On the classification of homeomorphisms of 2-manifolds and the classification of 3-manifolds,Acta Math. 142 (1979), 123–155.CrossRefMATHMathSciNetGoogle Scholar
  17. [Kau]
    L.H. Kauffman, State Models and the Jones Polynomial,Topology 26 (1987), 395–407.CrossRefMATHMathSciNetGoogle Scholar
  18. [Kir1]
    T.P. Kirkman, The enumeration, description and construction of knots of fewer than ten crossings,Trans. Roy. Soc. Edinburgh 32 (1885), 281–309.CrossRefMATHGoogle Scholar
  19. [Kir2]
    T.P. Kirkman, The 634 unifilar knots of ten crossings enumerated and defined,Trans. Roy. Soc. Edinburgh 32 (1885), 483–506.CrossRefMATHGoogle Scholar
  20. [KS]
    K. Kodama and M. Sakuma, Symmetry groups of prime knots up to 10 crossings, inKnots 90, Proceedings of the International Conference on Knot Theory and Related Topics, Osaka, Japan, 1990 (A. Kawauchi, ed.), Berlin: de Gruyter (1992), 323–340.Google Scholar
  21. [Lis]
    J.B. Listing, Vorstudien zur Topologie, Göttingen Studien, University of Göttingen, Germany (1848).Google Scholar
  22. [Lit 1]
    C.N. Little, On knots, with a census of order ten,Trans. Connecticut Acad. Sci. 18 (1885), 374–378.Google Scholar
  23. [Lit2]
    C.N. Little, Non alternate ± knots of orders eight and nine,Trans. Royal Soc. Edinburgh 35 (1889), 663–664.CrossRefMATHGoogle Scholar
  24. [Lit3]
    C.N. Little, Alternate ± knots of order 11,Trans. Roy. Soc. Edinburgh 36 (1890), 253–255.CrossRefMATHGoogle Scholar
  25. [Lit4]
    C.N. Little, Non-alternate ± knots,Trans. Royal Soc. Edinburgh 39 (1900), 771–778.CrossRefMATHGoogle Scholar
  26. [Lit5]
    C.N. Little, Knots, with a census for order ten, Ph.D. Thesis, Yale University (1885).Google Scholar
  27. [Men]
    W. Menasco, Closed incompressible surfaces in alternating knot and link complements,Topology 23(1) (1984), 37–44.CrossRefMATHMathSciNetGoogle Scholar
  28. [MP]
    K. Murasugi and J. Przytycki, The skein polynomial of a planar star product of two links,Math. Proc. Camb. Phil. Soc. 106(2) (1989), 273–276.CrossRefMATHMathSciNetGoogle Scholar
  29. [MT]
    W. Menasco and M.B. Thistlethwaite, The classification of alternating links,Ann. Math. 138 (1993), 113–171.CrossRefMATHMathSciNetGoogle Scholar
  30. [Mur1]
    K. Murasugi, The Jones polynomial and classical conjectures in knot theory,Topology 26 (1987), 187–194.CrossRefMATHMathSciNetGoogle Scholar
  31. [Mur2]
    K. Murasugi, Jones polynomials and classical conjectures in knot theory II,Math. Proc. Camb. Phil. Soc. 102 (1987), 317–318.CrossRefMATHMathSciNetGoogle Scholar
  32. [Perl]
    K. Perko, On the classification of knots,Proc. Am. Math. Soc. 46 (1974), 262–266.CrossRefMathSciNetGoogle Scholar
  33. [Per2]
    K. Perko, Invariants of 11-crossing knots, Prepublications Math. d’Orsay (1980)Google Scholar
  34. [Per3]
    K. Perko, Primality of certain knots,Topology Proc 7 (1982), 109–118.MATHMathSciNetGoogle Scholar
  35. [Pra]
    G. Prasad, Strong rigidity of Q-rank 1 lattices,Invent Math. 21 (1973), 255–286.CrossRefMATHMathSciNetGoogle Scholar
  36. [Rei]
    K. Reidemeister, Knotentheorie, Berlin: Springer-Verlag, 1932.Google Scholar
  37. [Ril]
    R. Riley, An elliptical path from parabolic representations to hyperbolic structures, inTopology of Low-Dimensional Manifolds, Proceedings, Sussex 1977 (R. Fenn, ed.), Springer Lecture Notes in Math. vol. 722, New York: Springer-Verlag (1979), 99–133.Google Scholar
  38. [SW]
    M. Sakuma and J. Weeks, The generalized tilt formula,Geometriae Dedicata 55 (1995), 115–123.CrossRefMATHMathSciNetGoogle Scholar
  39. [Sch]
    O. Schreier, Über die GruppenA aBb = 1,Abh. Math. Sem. Univ. Hamburg 3 (1924), 167–169.CrossRefMATHMathSciNetGoogle Scholar
  40. [ST]
    C. Sundberg and M. Thistlethwaite, The rate of growth of the number of prime alternating links and tangles,Pacific J. Math. 182 no.2 (1998), 329–358.CrossRefMATHMathSciNetGoogle Scholar
  41. [Tai]
    P.G. Tait, On knots I, II, III,Scientific Papers, Vol. I, Cambridge: Cambridge University Press (1898), 273–347.Google Scholar
  42. [Thi1]
    M.B. Thistlethwaite, Knot tabulations and related topjcs,Aspects of Topology in Memory of Hugh Dowker 1912–1982 (James and Kronheimer, eds.), Cambridge, England: Cambridge Press, London Math. Soc. Lecture Note Series 93, 1–76.Google Scholar
  43. [Thi2]
    M.B. Thistlethwaite, A spanning tree expansion of the Jones polynomial,Topology 26(3) (1987), 297–309.CrossRefMATHMathSciNetGoogle Scholar
  44. [Thi3]
    M.B. Thistlethwaite, Kauffman’s polynomial and alternating links,Topology 27(3) (1988), 311–318.CrossRefMATHMathSciNetGoogle Scholar
  45. [Thi4]
    M.B. Thistlethwaite, On the structure and paucity of alternating tangles and links, preprint.Google Scholar
  46. [Tro]
    H.F. Trotter, Noninvertible knots exist,Topology 2 (1964), 275–280.CrossRefMathSciNetGoogle Scholar
  47. [Wks]
    J. Weeks, Convex hulls and isometries of cusped hyperbolic 3-manifolds,Topol. Appl. 52 (1993), 127–149.CrossRefMATHMathSciNetGoogle Scholar
  48. [Wei]
    D.J.A. Welsh, On the number of knots and links,Colloq Math. Soc. J. Bolyai 60 (1991), 713–718.MathSciNetGoogle Scholar
  49. [Wil]
    R.F. Williams, The braid index of an algebraic link,Braids (Santa Cruz, CA, 1986), Contemporary Mathematics Series Vol. 78, Providence, Rl: American Mathematical Society (1988).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 1998

Authors and Affiliations

  1. 1.Pitzer CollegeClaremontUSA
  2. 2.University of TennesseeKnoxvilleUSA
  3. 3.CantonNYUSA

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