We give a simple naming argument for establishing lower bounds on the
combinatorial distance between (positive) braid words

Dehornoy, Patrick

English

arXiv

Abstract. We give a simple naming argument for establishing lower bounds on the combinatorial distance between (positive) braid words. 1 It is well-known that, for n&gt; 3, Artin’s braid group Bn, which is the group defined by the presentationø σ1,...,σn−1 ØØØØ σiσj = σjσi for |i − j |&gt; 2σiσjσi = σjσiσj for |i − j |  = 1 has a quadratic Dehn function, i.e., there exist constants Cn, C0n such that, if w is an n-strand braid word of length ` that represents the unit braid, then the number of braid relations needed to transform w into the empty word is at most Cn`2 and, on the other hand, there exists for each ` at least one length ` word w such that the minimal number of such braid relations is at least C0n`2—see for instance [2]. In a recent posting [3], Hass, Kalka, and Nowik developed a knot theoretical argument for establishing lower bounds on the combinatorial distance between two equivalent positive braid words, i.e., on the minimal number of braid relations needed to transform the former into the latter. Using some knot invariants intro

Patrick Dehornoy

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COMBINATORIAL DISTANCE BETWEEN BRAID WORDS

We give a simple naming argument for establishing lower bounds on the combinatorial distance between (positive) braid words

HAL - Normandie Université

Combinatorial distance between braid words

http://www.math.unicaen.fr/%7Edehornoy/Papers/Dhz.pdf

Combinatorial distance between braid words

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