Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Garside theory and subsurfaces: Some examples in braid groups

Garside theory and subsurfaces: Some examples in braid groups AbstractGarside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid braids with a fixed number of strands, the size of this set is bounded by a polynomial in the length of the braids. In this paper we suggest a more precise bound: for rigid braids with N strands and of Garside length L, the sliding circuit set should have at most C⋅LN-2{C\cdot L^{N-2}} elements, for some constant C. We construct a family of braids which realise this potential worst case.Our example braids suggest that having a large sliding circuit set is a geometric property of braids, as our examples have multiple subsurfaces with large subsurface projection; thus they are “almost reducible” in multiple ways, and act on the curve graph with small translation distance. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups Complexity Cryptology de Gruyter

Garside theory and subsurfaces: Some examples in braid groups

Groups Complexity Cryptology , Volume 11 (2): 15 – Nov 1, 2019

Loading next page...
 
/lp/de-gruyter/garside-theory-and-subsurfaces-some-examples-in-braid-groups-5fWWkHdWYN
Publisher
de Gruyter
Copyright
© 2019 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1869-6104
eISSN
1869-6104
DOI
10.1515/gcc-2019-2007
Publisher site
See Article on Publisher Site

Abstract

AbstractGarside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid braids with a fixed number of strands, the size of this set is bounded by a polynomial in the length of the braids. In this paper we suggest a more precise bound: for rigid braids with N strands and of Garside length L, the sliding circuit set should have at most C⋅LN-2{C\cdot L^{N-2}} elements, for some constant C. We construct a family of braids which realise this potential worst case.Our example braids suggest that having a large sliding circuit set is a geometric property of braids, as our examples have multiple subsurfaces with large subsurface projection; thus they are “almost reducible” in multiple ways, and act on the curve graph with small translation distance.

Journal

Groups Complexity Cryptologyde Gruyter

Published: Nov 1, 2019

There are no references for this article.