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Garside theory and subsurfaces : some examples in braid groups
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Schleimer, Saul and Wiest, Bert (2019) Garside theory and subsurfaces : some examples in braid groups. Groups Complexity Cryptology, 11 (2). pp. 61-75. doi:10.1515/gcc-2019-2007 ISSN 1867-1144.
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Official URL: https://doi.org/10.1515/gcc-2019-2007
Abstract
Garside-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 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.
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Braid theory, Conjugacy classes , Group theory | ||||||||
| Journal or Publication Title: | Groups Complexity Cryptology | ||||||||
| Publisher: | Walter de Gruyter GmbH | ||||||||
| ISSN: | 1867-1144 | ||||||||
| Official Date: | 1 November 2019 | ||||||||
| Dates: |
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| Volume: | 11 | ||||||||
| Number: | 2 | ||||||||
| Page Range: | pp. 61-75 | ||||||||
| DOI: | 10.1515/gcc-2019-2007 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Date of first compliant deposit: | 30 October 2019 | ||||||||
| Date of first compliant Open Access: | 23 October 2020 |
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