The infimum, supremum, and geodesic length of a braid conjugacy class

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Algorithmic solutions to the conjugacy problem in the braid groups B-n, n = 2, 3, 4,... were given in earlier work. This note concerns the computation of two integer class invariants, known as "inf" and "sup." A key issue in both algorithms is the number m of times one must "cycle" (resp. "decycle") in order to either increase inf (resp. decrease sup) or to be sure that it is already maximal (resp. minimal) for the class. Our main result is to prove that m is bounded above by ((n(2) - n)/2) - I in the situation stated by E. A. Elrifai and H. R. Morton (1994, Quart. J. Math. Oxford 45, 479-497) and by n - 2 in the situation stated by authors (1998, Adv. Math. 139, 322-353). It follows immediately that the computation of inf and sup is polynomial in both word length and braid index, in both algorithms. The integers inf and sup determine (but are not determined by) the shortest geodesic length for elements in a conjugacy class, and so we also obtain a polynomial-time algorithm for computing this length. (C) 2001 Elsevier Science.
Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
Issue Date
2001-12
Language
English
Article Type
Article
Citation

ADVANCES IN MATHEMATICS, v.164, no.1, pp.41 - 56

ISSN
0001-8708
URI
http://hdl.handle.net/10203/78699
Appears in Collection
MA-Journal Papers(저널논문)
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