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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].

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)