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 $13^-$ to $11^-$, where we fail to consider the case: $V_2 = 4, V_4 > 0, V_j=0$ if $j not= 2,4,$ 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