Mathematics of Computation

Author(s): A. Stoimenow.
Journal: Math. Comp. 72 (2003), 2043-2057.
MSC (2000): Primary 57M25
Posted: March 26, 2003
Retrieve article in: PDF DVI PostScript

Abstract: Answering negatively a question of Bleiler, we give examples of knots where the difference between minimal and maximal unknotting number of minimal crossing number diagrams grows beyond any extent.

[Ad]: C. C. Adams, Das Knotenbuch, Spektrum Akademischer Verlag, Berlin, 1995 (The knot book, W. H. Freeman & Co., New York, 1994). MR 94m:57007
[Bl]: S. A. Bleiler, A note on unknotting number, Math. Proc. Camb. Phil. Soc. 96 (1984), 469-471. MR 85m:57006
[Bl2]: -, Realizing concordant polynomials with prime knots, Pacific J. Math. 100(2) (1982), 249-257. MR 84e:57004
[BW]: M. Boileau and C. Weber, Le problème de J. Milnor sur le nombre gordien des n $\oe$ uds algébriques, Enseign. Math. 30 (1984), 173-222. MR 86c:57004
[Co]: J. H. Conway, An enumeration of knots and links, in ``Computational problems in abstract algebra'' (J. Leech, ed.), 329-358, Pergamon Press, 1969. MR 41:2661
[Cr]: P. R. Cromwell, Homogeneous links, J. London Math. Soc. (series 2) 39 (1989), 535-552. MR 90f:57001
[FW]: J. Franks and R. F. Williams, Braids and the Jones polynomial, Trans. Amer. Math. Soc. 303 (1987), 97-108. MR 88k:57006
[Ga]: D. Gabai, Detecting fibred links in , Comment. Math. Helv. 61(4) (1986), 519-555. MR 88c:57009
[H]: P. Freyd, J. Hoste, W. B. R. Lickorish, K. Millett, A. Ocneanu and D. Yetter, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985), 239-246. MR 86e:57007
[HT]: J. Hoste and M. Thistlethwaite, KnotScape, a knot polynomial calculation program, available at http://www.math.utk.edu/morwen.
[HTW]: J. Hoste, M. Thistlethwaite, and J. Weeks, The first 1,701,936 knots, Math. Intell. 20 (4) (1998), 33-48. MR 97i:57015
[K]: T. Kanenobu, Relations between the Jones and Q polynomials of 2-bridge and 3-braid links, Math. Ann. 285 (1989), 115-124. MR 90i:57002
[KM]: - and H. Murakami, 2-bridge knots of unknotting number one, Proc. Amer. Math. Soc. 98(3) (1986), 499-502. MR 87i:57005
[Ka]: L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), 417-471. MR 90g:57007
[Ka2]: -, State models and the Jones polynomial, Topology 26 (1987), 395-407 MR 88f:57006.
[Ki]: M. Kidwell, On the degree of the Brandt-Lickorish-Millett-Ho polynomial of a link, Proc. Amer. Math. Soc. 100 (1987), 755-761. MR 89b:57003
[KL]: R. C. Kirby and W. B. R. Lickorish, Prime knots and concordance, Math. Proc. Cambridge Philos. Soc. 86(3) (1979), 437-441. MR 80k:57011
[KMr]: P. B. Kronheimer and T. Mrowka, The genus of embedded surfaces in in the projective plane, Math. Res. Lett. 1 (1994), 797-808. MR 96a:57073
[Li]: W. B. R. Lickorish, The unknotting number of a classical knot, in ``Contemporary Mathematics'' 44 (1985), 117-119. MR 87a:57012
[LM]: -and K. C. Millett, A polynomial invariant for oriented links, Topology 26 (1) (1987), 107-141. MR 88b:57012
[Me]: W. W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1) (1984), 37-44. MR 86b:57004
[MT]: - and M. B. Thistlethwaite, The Tait flyping conjecture, Bull. Amer. Math. Soc. 25 (2) (1991), 403-412. MR 92b:57017
[Mo]: H. R. Morton, Seifert circles and knot polynomials, Proc. Camb. Phil. Soc. 99 (1986), 107-109. MR 87c:57006
[MS]: - and H. Short, The 2-variable polynomial of cable knots, Math. Proc. Camb. Philos. Soc. 101 (1987), 267-278. MR 88f:57009
[Mu]: K. Murasugi, Jones polynomial and classical conjectures in knot theory, Topology 26 (1987), 187-194. MR 88m:57010
[Mu2]: -, On the braid index of alternating links, Trans. Amer. Math. Soc. 326 (1) (1991), 237-260. MR 91j:57009
[MP]: -and J. Przytycki, The skein polynomial of a planar star product of two links, Math. Proc. Cambridge Philos. Soc. 106(2) (1989), 273-276. MR 90f:57008
[Na]: Y. Nakanishi, Unknotting numbers and knot diagrams with the minimum crossings, Math. Seminar Notes Kobe Univ. 11(2) (1983), 257-258. MR 85h:57008
[Ro]: D. Rolfsen, Knots and links, Publish or Parish, 1976. MR 58:24236
[S]: J. R. Stallings, Constructions of fibred knots and links, ``Algebraic and geometric topology'' Proc. Sympos. Pure Math., vol. 32, part 2, Amer. Math. Soc., Providence, RI, 1978, pp. 55-60. MR 80e:57004
[St]: A. Stoimenow, Some examples related to 4-genera, unknotting numbers, and knot polynomials, Jour. London Math. Soc. 63(2) (2001), 487-500. MR 2002c:57011
[St2]: -, The granny and the square tangle and the unknotting number, Topol. Appl. 117 (2002), 59-75.
[St3]: -, On the crossing number of positive knots and braids and braid index criteria of Jones and Morton-Williams-Franks, math.GT/0110016, Trans. Amer. Math. Soc., 354 (2002), 3927-3924.
[Th]: M. B. Thistlethwaite, A spanning tree expansion for the Jones polynomial, Topology 26 (1987), 297-309. MR 88h:57007
[Th2]: -, Kauffman's polynomial and alternating links, Topology 27(3) (1988), 311-318. MR 90c:57005
[Th3]: -, On the Kauffman polynomial of an adequate link, Invent. Math. 93(2) (1988), 285-296. MR 89g:57009
[Th4]: -, On the structure and scarcity of alternating links and tangles, J. Knot Theory Ramifications 7(7) (1998), 981-1004. MR 99k:57031
[Tr]: P. Traczyk, A criterion for signed unknotting number, Contemporary Mathematics 233 (1999), 215-220. MR 2000e:57016
[Va]: Quach thi Câm Vân, On a theorem on partially summing tangles by Lickorish, Math. Proc. Cambridge Philos. Soc. 93(1) (1983), 63-66. MR 84c:57003
[We]: H. Wendt, Die Gordische Auflösung von Knoten, Math. Z. 42 (1937), 680-696.

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