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Transactions of the American Mathematical Society
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
Published electronically: June 10, 2002
MathSciNet review: 1926860
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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.


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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: http://dx.doi.org/10.1090/S0002-9947-02-03022-2
PII: S 0002-9947(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