2,734 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
On an action of the braid group B_{2g+2} on the free group F_{2g}
We construct an action of the braid group B_{2g+2} on the free group F_{2g}
extending an action of B_4 on F_2 introduced earlier by Reutenauer and the
author. Our action induces a homomorphism from B_{2g+2} into the symplectic
modular group Sp_{2g}(Z). In the special case g=2 we show that the latter
homomorphism is surjective and determine its kernel, thus obtaining a
braid-like presentation of Sp_4(Z).Comment: 11 pages. Minor changes in v
Mapping class group and U(1) Chern-Simons theory on closed orientable surfaces
U(1) Chern-Simons theory is quantized canonically on manifolds of the form
, where is a closed orientable surface. In
particular, we investigate the role of mapping class group of in the
process of quantization. We show that, by requiring the quantum states to form
representation of the holonomy group and the large gauge transformation group,
both of which are deformed by quantum effect, the mapping class group can be
consistently represented, provided the Chern-Simons parameter satisfies an
interesting quantization condition. The representations of all the discrete
groups are unique, up to an arbitrary sub-representation of the mapping class
group. Also, we find a duality of the representations.Comment: 17 pages, 3 figure
A new algorithm for recognizing the unknot
The topological underpinnings are presented for a new algorithm which answers
the question: `Is a given knot the unknot?' The algorithm uses the braid
foliation technology of Bennequin and of Birman and Menasco. The approach is to
consider the knot as a closed braid, and to use the fact that a knot is
unknotted if and only if it is the boundary of a disc with a combinatorial
foliation. The main problems which are solved in this paper are: how to
systematically enumerate combinatorial braid foliations of a disc; how to
verify whether a combinatorial foliation can be realized by an embedded disc;
how to find a word in the the braid group whose conjugacy class represents the
boundary of the embedded disc; how to check whether the given knot is isotopic
to one of the enumerated examples; and finally, how to know when we can stop
checking and be sure that our example is not the unknot.Comment: 46 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTVol2/paper9.abs.htm
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
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