784 research outputs found
Quantum Algorithms for the Jones Polynomial
This paper gives a generalization of the AJL algorithm and unitary braid
group representation for quantum computation of the Jones polynomial to
continuous ranges of values on the unit circle of the Jones parameter. We show
that our 3-strand algorithm for the Jones polynomial is a special case of this
generalization of the AJL algorithm. The present paper uses diagrammatic
techniques to prove these results. The techniques of this paper have been used
and will be used in the future in work with R. Marx, A. Fahmy, L. H. Kauffman,
S. J. Lomonaco Jr.,A. Sporl, N. Pomplun, T. Schulte Herbruggen, J. M. Meyers,
and S. J. Glaser on NMR quantum computation of the Jones polynomial.Comment: 11 pages, 4 figures, LaTeX documen
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
On the Sato-Tate conjecture for non-generic abelian surfaces
We prove many non-generic cases of the Sato-Tate conjecture for abelian
surfaces as formulated by Fite, Kedlaya, Rotger and Sutherland, using the
potential automorphy theorems of Barnet-Lamb, Gee, Geraghty and Taylor.Comment: 21 pages. Minor changes and corrections. With an appendix by Francesc
Fit\'e. Essentially final version, to appear in Transactions of the AM
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