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On reduction curves and Garside properties of braids. (English) Zbl 1255.20035
Cogolludo-Agustín, José Ignacio (ed.) et al., Topology of algebraic varieties and singularities. Invited papers of the conference in honor of Anatoly Libgober’s 60th birthday, Jaca, Spain, June 22–26, 2009. Providence, RI: American Mathematical Society (AMS); Madrid: Real Sociedad Matemática Española (ISBN 978-0-8218-4890-6/pbk). Contemporary Mathematics 538, 227-244 (2011).
Given any braid $$x\in B_n$$, the set of sliding circuits $$SC(x)$$ has been used by V. Gebhardt and J. González-Meneses, [J. Symb. Comput. 45, No. 6, 629-656 (2010; Zbl 1235.20032)], in order to simplify the solution of the conjugacy problem in $$B_n$$. $$SC(x)$$ consists of all conjugates $$y$$ of $$x$$ that belong to a sliding circuit, meaning that $$y$$ coincides with the cyclic sliding $$\mathfrak s^m(y)$$ for some $$m>0$$. The cyclic sliding $$\mathfrak s(y)$$ is defined as the conjugate $$\mathfrak p(y)^{-1}y\mathfrak p(y)$$, where $$\mathfrak p(y)$$ is the preferred prefix of $$y$$ [V. Gebhardt and J. González-Meneses, Math. Z. 265, No. 1, 85-114 (2010; Zbl 1253.20034)].
The paper under review is aimed to provide a family of braids $$\beta$$ whose number of sliding circuits is exponential in both the number of strands and the canonical length. Namely, for $$n\geq 3$$ and $$k\geq 1$$ the braid $$\beta=(\sigma_1\cdots\sigma_{n-1})^{nk+1}\sigma_{n+1}^{4k+1}\in B_{n+2}$$ of canonical length $$\ell(\beta)=4k+1$$ is proved to have $$\#(SC(\beta))\geq 2^{n-2}(n-1)^{2k-1}$$.
This result is proved by considering the decomposition of braids along standard canonical reduction systems, that is invariant sets of disjoint essential round circles.
For the entire collection see [Zbl 1210.14004].

##### MSC:
 20F36 Braid groups; Artin groups 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 57M25 Knots and links in the $$3$$-sphere (MSC2010)
##### Citations:
Zbl 1235.20032; Zbl 1253.20034