Abstract
For the proof of Theorem 1.3 we will need to study the behavior of the bound (2.11) in the Morton-Franks-Williams inequality (we abbreviate as MFW) on positive braids. This was begun by Nakamura [Na], who settled the case MFW = 2 in the suggestive way: such braids represent only the (2, n)-torus links. (The case MFW = 1 is trivial.) We will introduce a method that considerably simplifies his proof (but still makes use of some of his ideas), and then go on to deal with MFW = 3. The example of non-sharp MFW inequality, 139365 in [HT] (the connected 2-cable of the trefoil), given in [MS], is in fact only among a small family of exceptional cases.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Notice we use n in a different meaning from before!
References
J.S. Birman, H. Wenzl, Braids, link polynomials and a new algebra. Trans. Am. Math. Soc. 313(1), 249–273 (1989)
J. Franks, R.F. Williams, Braids and the Jones-Conway polynomial. Trans. Am. Math. Soc. 303, 97–108 (1987)
J. Hoste, M. Thistlethwaite, KnotScape, a knot polynomial calculation and table access program. Available at http://www.math.utk.edu/~morwen
J. Hoste, M. Thistlethwaite, J. Weeks, The first 1,701,936 knots. Math. Intell. 20(4), 33–48 (1998)
M. Ishikawa, On the Thurston-Bennequin invariant of graph divide links. Math. Proc. Camb. Philos. Soc. 139(3), 487–495 (2005)
W.B.R. Lickorish, M.B. Thistlethwaite, Some links with non-trivial polynomials and their crossing numbers. Comment. Math. Helv. 63, 527–539 (1988)
J. Murakami, The Kauffman polynomial of links and representation theory. Osaka J. Math. 24(4), 745–758 (1987)
H.R. Morton, H.B. Short, The2-variable polynomial of cable knots. Math. Proc. Camb. Philos. Soc. 101(2), 267–278 (1987)
K. Murasugi, On the braid index of alternating links. Trans. Am. Math. Soc. 326(1), 237–260 (1991)
T. Nakamura, Notes on the braid index of closed positive braids. Topology Appl. 135(1–3), 13–31 (2004)
A. Stoimenow, On the crossing number of positive knots and braids and braid index criteria of Jones and Morton-Williams-Franks. Trans. Am. Math. Soc. 354(10), 3927–3954 (2002)
A. Stoimenow, Coefficients and non-triviality of the Jones polynomial, J. Reine Angew. Math. 657 (2011), 1–55; see also math.GT/0606255
A. Stoimenow, The braid index and the growth of Vassiliev invariants. J. Knot Theory Ram. 8(6), 799–813 (1999)
M. B. Thistlethwaite, On the Kauffman polynomial of an adequate link. Invent. Math. 93(2), 285–296 (1988)
Y. Yokota, Polynomial invariants of positive links. Topology 31(4), 805–811 (1992)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 The Author(s)
About this chapter
Cite this chapter
Stoimenow, A. (2017). Positivity of 3-Braid Links. In: Properties of Closed 3-Braids and Braid Representations of Links . SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-68149-8_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-68149-8_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68148-1
Online ISBN: 978-3-319-68149-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)