101 research outputs found
Braids, knots and contact structures
These notes were prepared to supplement the talk that I gave on Feb 19, 2004,
at the First East Asian School of Knots and Related Topics, Seoul, South Korea.
In this article I review aspects of the interconnections between braids, knots
and contact structures on Euclidean 3-space. I discuss my recent work with
William Menasco (arXiv math.GT/0310279)} and (arXiv math.GT/0310280). In the
latter we prove that there are distinct transversal knot types in contact
3-space having the same topological knot type and the same Bennequin invariant.Comment: 10 pages, 5 figure
Erratum: Studying Links via Closed Braids IV: Composite Links and Split Links
The purpose of this erratum is to fill a gap in the proof of the `Composite
Braid Theorem' in the manuscript "Studying Links Via Closed Braids IV:
Composite Links and Split Links (SLVCB-IV)", Inventiones Math, \{bf 102\} Fasc.
1 (1990), 115-139. The statement of the theorem is unchanged. The gap occurs on
page 135, lines to , where we fail to consider the case: if and all 4 vertices of valence 2 are bad.
At the end of this Erratum we make some brief remarks on the literature, as
it has evolved during the 14 years between the publication of SLVCB-IV and the
submission of this Erratum.Comment: 6 pages, 4 figures. This is an Erratum to "Studying Links Vai Closed
Braids IV: Composite Links and Split Links", Inventiones Math., 102 Facs. 1
(190), 115-13
Braids: A Survey
This article is about Artin's braid group and its role in knot theory. We set
ourselves two goals: (i) to provide enough of the essential background so that
our review would be accessible to graduate students, and (ii) to focus on those
parts of the subject in which major progress was made, or interesting new
proofs of known results were discovered, during the past 20 years. A central
theme that we try to develop is to show ways in which structure first
discovered in the braid groups generalizes to structure in Garside groups,
Artin groups and surface mapping class groups. However, the literature is
extensive, and for reasons of space our coverage necessarily omits many very
interesting developments. Open problems are noted and so-labelled, as we
encounter them.Comment: Final version, revised to take account of the comments of readers. A
review article, to appear in the Handbook of Knot Theory, edited by W.
Menasco and M. Thistlethwaite. 91 pages, 24 figure
Stabilization in the braid groups II: Transversal simplicity of knots
The main result of this paper is a negative answer to the question: are all
transversal knot types transversally simple? An explicit infinite family of
examples is given of closed 3-braids that define transversal knot types that
are not transversally simple. The method of proof is topological and indirect.Comment: This is the version published by Geometry & Topology on 4 October
2006. Part I (arXiv:math/0310279) is also published in this volum
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