Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the crossing number of positive knots and braids and braid index criteria of Jones and Morton-Williams-Franks


Author: A. Stoimenow
Journal: Trans. Amer. Math. Soc. 354 (2002), 3927-3954
MSC (2000): Primary 57M25; Secondary 20F10, 20F36
DOI: https://doi.org/10.1090/S0002-9947-02-03022-2
Published electronically: June 10, 2002
MathSciNet review: 1926860
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give examples of knots with some unusual properties of the crossing number of positive diagrams or strand number of positive braid representations. In particular, we show that positive braid knots may not have positive minimal (strand number) braid representations, giving a counterpart to results of Franks-Williams and Murasugi. Other examples answer questions of Cromwell on homogeneous and (partially) of Adams on almost alternating knots.

We give a counterexample to, and a corrected version of, a theorem of Jones on the Alexander polynomial of 4-braid knots. We also give an example of a knot on which all previously applied braid index criteria fail to estimate sharply (from below) the braid index. A relation between (generalizations of) such examples and a conjecture of Jones that a minimal braid representation has unique writhe is discussed.

Finally, we give a counterexample to Morton's conjecture relating the genus and degree of the skein polynomial.


References [Enhancements On Off] (What's this?)

  • 1. C. C. Adams, Das Knotenbuch, Spektrum Akademischer Verlag, Berlin, 1995 (The knot book, W. H. Freeman and Co., New York, 1994). MR 94m:57007
  • 2. C. C. Adams et al., Almost alternating links, Topol. Appl. 46 (1992), 151-165. MR 93h:57005
  • 3. J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), 275-306.
  • 4. C. Aneziris, The mystery of knots. Computer programming for knot tabulation. Series on Knots and Everything 20, World Scientific, 1999.
  • 5. D. Bennequin, Entrelacements et équations de Pfaff, Soc. Math. de France, Astérisque 107-108 (1983), 87-161. MR 86e:58070
  • 6. J. S. Birman, On the Jones polynomial of closed 3-braids, Invent. Math. 81 (1985), 287-294. MR 86m:57006
  • 7. J. S. Birman and W. W. Menasco, Studying knots via braids VI: A non-finiteness theorem, Pacific J. Math. 156 (1992), 265-285. MR 93m:57005
  • 8. -, Studying knots via braids III: Classifying knots which are closed 3 braids, Pacific J. Math. 161(1993), 25-113. MR 94i:57005
  • 9. J. S. Birman and R. F. Williams, Knotted periodic orbits in dynamical systems - I, Lorenz's equations, Topology 22(1) (1983), 47-82. MR 84k:58138
  • 10. R. D. Brandt, W. B. R. Lickorish and K. Millett, A polynomial invariant for unoriented knots and links, Invent. Math. 74 (1986), 563-573. MR 87m:57003
  • 11. J. van Buskirk, Positive links have positive Conway polynomial, Springer Lecture Notes in Math. 1144 (1983), 146-159. MR 87f:57007
  • 12. L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fibonacci Quart. 11 (1973), 337-386. MR 48:10969
  • 13. 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
  • 14. P. R. Cromwell, Homogeneous links, J. London Math. Soc. (series 2) 39 (1989), 535-552. MR 90f:57001
  • 15. -, Positive braids are visually prime, Proc. London Math. Soc. 67 (1993), 384-424. MR 95c:57008
  • 16. -, A note on Morton's conjecture concerning the lowest degree of a $2$-variable knot polynomial, Pacific J. Math. 160(2) (1993), 201-205. MR 94h:57008
  • 17. P. R. Cromwell and H. R. Morton, Positivity of knot polynomials on positive links, J. Knot Theory Ramifications 1 (1992), 203-206. MR 93i:57005
  • 18. O. T. Dasbach and B. Mangum, On McMullen's and other inequalities for the Thurston norm of link complements, Algebraic and Geometric Topology, 1 (2001), 321-347.
  • 19. C. H. Dowker and M. B. Thistlethwaite, Classification of knot projections, Topol. Appl. 16 (1983), 19-31. MR 85e:57003
  • 20. Th. Fiedler, A small state sum for knots, Topology 32 (2) (1993), 281-294. MR 94c:57006
  • 21. J. Franks and R. F. Williams, Braids and the Jones polynomial, Trans. Amer. Math. Soc. 303 (1987), 97-108. MR 88k:57006
  • 22. D. Gabai, Foliations and genera of links, Topology 23 (1984), 381-394. MR 86h:57006
  • 23. F. Hirzebruch, Singularities and exotic spheres, Seminaire Bourbaki 10, Exp. No. 314, 13-32, Soc. Math. France, Paris, 1995. MR 99f:00042
  • 24. 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
  • 25. C. F. Ho, A polynomial invariant for knots and links - preliminary report, Abstracts Amer. Math. Soc. 6 (1985), 300.
  • 26. J. Hoste and M. Thistlethwaite, KnotScape, a knot polynomial calculation and table access program, available at http://www.math.utk.edu/~morwen.
  • 27. V. F. R. Jones, A polynomial invariant of knots and links via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985), 103-111. MR 86e:57006
  • 28. -, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987), 335-388. MR 89c:46092
  • 29. 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
  • 30. M. Kidwell and A. Stoimenow, Examples Relating to the Crossing Number, Writhe, and Maximal Bridge Length of Knot Diagrams, Michigan Math. J. (to appear).
  • 31. M. Kobayashi and T. Kobayashi, On canonical genus and free genus of knot, J. Knot Theory Ramifications 5(1) (1996), 77-85. MR 97d:57008
  • 32. W. B. R. Lickorish and K. C. Millett, A polynomial invariant for oriented links, Topology 26 (1) (1987), 107-141. MR 88b:57012
  • 33. W. W. Menasco and M. B. Thistlethwaite, The Tait flyping conjecture, Bull. Amer. Math. Soc. 25 (2) (1991), 403-412. MR 92b:57017
  • 34. H. R. Morton, Seifert circles and knot polynomials, Math. Proc. Cambridge Philos. Soc. 99 (1986), 107-109. MR 87c:57006
  • 35. -, The Burau matrix and Fiedler's invariant for a closed braid, Topol. Appl. 95(3) (1999), 251-256. MR 2001i:57025
  • 36. -, Threading knot diagrams, Math. Proc. Cambridge Philos. Soc. 99(2) (1986), 247-260. MR 87c:57007
  • 37. -(ed.), Problems, in ``Braids'', Santa Cruz, 1986 (J. S. Birman and A. L. Libgober, eds.), Contemp. Math. 78, 557-574. MR 89f:20003
  • 38. -(ed.), Problems, in the Proceedings of the International Conference on Knot Theory ``Knots in Hellas, 98'', Series on Knots and Everything 24, World Scientific, 2000, 547-559.
  • 39. H. R. Morton and E. A. El-Rifai, Algorithms for positive braids, Quart. J. Math. Oxford Ser. 2, 45(180) (1994), 479-497. MR 96b:20052
  • 40. H. R. Morton and H. B. Short, Calculating the $2$-variable polynomial for knots presented as closed braids, J. Algorithms 11(1) (1990), 117-131.MR 91f:57004
  • 41. -, The $2$-variable polynomial of cable knots, Math. Proc. Cambridge Philos. Soc. 101(2) (1987), 267-278. MR 88f:57009
  • 42. J. Murakami, The Kauffman polynomial of links and representation theory, Osaka J. Math. 24(4) (1987), 745-758. MR 89c:57007
  • 43. K. Murasugi, On the braid index of alternating links, Trans. Amer. Math. Soc. 326 (1) (1991), 237-260. MR 91j:57009
  • 44. -, Jones polynomials and classical conjectures in knot theory, Topology 26 (1987), 187-194. MR 88m:57010
  • 45. -, On closed 3-braids, Memoirs Amer. Math. Soc. 151 (1974), American Mathematical Society, Providence, RI. MR 50:8496
  • 46. K. Murasugi 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
  • 47. T. Nakamura, Positive alternating links are positively alternating, J. Knot Theory Ramifications 9(1) (2000), 107-112. MR 2001a:57016
  • 48. D. Rolfsen, Knots and links, Publish or Perish, 1976. MR 58:24236
  • 49. L. Rudolph, Nontrivial positive braids have positive signature, Topology 21(3) (1982), 325-327. MR 83c:57009
  • 50. -, Braided surfaces and Seifert ribbons for closed braids, Comment. Math. Helv. 58 (1983), 1-37. MR 84j:57006
  • 51. N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, accessible on the Internet address http://www.research.att.com/~njas/sequences.
  • 52. J. R. Stallings, Constructions of fibred knots and links, Algebraic and Geometric Topology (Stanford, CA, 1996), Proc. Sympos. Pure Math., vol. 32, part 2, Amer. Math. Soc., Providence, RI, 1978, pp. 55-60. MR 80e:57004
  • 53. A. Stoimenow, Positive knots, closed braids and the Jones polynomial, preprint math/9805078.
  • 54. -, Knots of genus two, preprint.
  • 55. -, Generating functions, Fibonacci numbers, and rational knots, preprint.
  • 56. -, Knots of genus one, Proc. Amer. Math. Soc. 129(7) (2001), 2141-2156. MR 2002c:57012
  • 57. -, Some inequalities between knot invariants, accepted by Internat. J. Math.
  • 58. P. Vogel, Representation of links by braids: A new algorithm, Comment. Math. Helv. 65 (1990), 104-113. MR 90k:57013

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57M25, 20F10, 20F36

Retrieve articles in all journals with MSC (2000): 57M25, 20F10, 20F36


Additional Information

A. Stoimenow
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany
Address at time of publication: Department of Mathematics, University of Toronto, Toronto, Canada M5S 3G3
Email: alex@mpim-bonn.mpg.de, stoimeno@math.toronto.edu

DOI: https://doi.org/10.1090/S0002-9947-02-03022-2
Received by editor(s): November 10, 2001
Received by editor(s) in revised form: February 12, 2002
Published electronically: June 10, 2002
Additional Notes: Supported by a DFG postdoc grant.
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society