A family of pseudo-Anosov braids with small dilatation

By E. Hironaka, Eiko Kin. 2005

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1mCn : Matsuoka’s example [22] appears as 1;1 , and Ko, Los and Song’s example [18] as 2;1 . For any m;n 1, m;n is pseudo-Anosov (Theorem 3.9). The dilatations of m;m coincide with those found by Brinkmann [7] (see also Section 4.2), who also shows that the dilatations arising in this family can be made arbitrarily close to 1. It turns out that one may find smaller dilatations by passing a strand of m;n once around the remaining strands. As a particular example, we consider the braids m;n defined by taking the rightmost-strand of m;n and passing it counter-clockwise once around the remaining strands. Figure 1 gives an illustration of m;n and m;n . The braid 1;3 is conjugate to Ham and Song’s braid 1 2 3 4 1 2 . Forjm nj 1, we show that m;n is periodic or reducible. Otherwise m;n is pseudo-Anosov with

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0b730d29aacb9f8ae6f4813e1364e662af6aad58

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Citations in the corpus

The asymptotic behavior of least pseudo-Anosov dilatations (2008, methodology): “and for closed surfaces with genus g 2, log 2 12g 12 lg;0 log 11 g : The bounds on lg;0 have been improved by a number of authors; see Bauer [1], McMullen [10], Minakawa [11]and Hironaka and Kin [7].”
The lower central series and pseudo-Anosov dilatations (2006, background): “The best known general upper bound is log(2 + √ 3)/g given by Hironaka–Kin [ HK ] and Minakawa [Mk].”