On Word Reversing in Braid Groups

It has been conjectured that in a braid group, or more generally in a Garside group, applying any sequence of monotone equivalences and word reversings can increase the length of a word by at most a linear factor depending on the group presentation only. We give a counter-example to this conjecture, but, on the other hand, we establish length upper bounds for the case when only right reversing is involved. We also state a new conjecture which would, like the above one, imply that the space complexity of the handle reduction algorithm is linear. This paper was motivated by attempts to estimate the complexity of the handle reduction algorithm in braid groups [4], via a detailed study of word reversings. Word reversing is a general combinatorial method for investigating monoids and groups specified by explicit presentations [3, 6, 8]. In good cases, typically in the case of braid groups [3] and, more generally, Garside groups [7], it provides algorithmic solutions to the word problem, as ...

• Type: Journal
• Year: 2006
• Where: IJAC
• Authors: Patrick Dehornoy, Bert Wiest
• Keywords: Braid Groups, Handle Reduction Algorithm, IJAC 2006, Word Reversings