2005/06/30 by Eon-Kyung Lee, Sangjin Lee, Sang-Jin Lee
Mathematics · #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT #msc:20F10 #msc:20F36
paper · pdf · doi:10.1016/j.jalgebra.2008.03.033
published as Journal of Algebra, vol. 320, no. 2. pp.783-820, 2008 · 38 pages, 18 figures, published version
openalex publication_date 2008/05/17 · arxiv created 2008/07/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
Let Dn denote the n-punctured disk in the complex plane, where the punctures are on the real axis. An n-braid α is said to be reducible if there exists an essential curve system \C in Dn, called a reduction system of α, such that α*\C=\C where α*\C denotes the action of the braid α on the curve system \C. A curve system \C in Dn is said to be standard if each of its components is isotopic to a round circle centered at the real axis. In this paper, we study the characteristics of the braids sending a curve system to a standard curve system, and then the characteristics of the conjugacy classes of reducible braids. For an essential curve system \C in Dn, we define the standardizer of \C as \St(\C)=\P∈ Bn+:P*\Cis standard\ and show that \St(\C) is a sublattice of Bn+. In particular, there exists a unique minimal element in \St(\C). Exploiting the minimal elements of standardizers together with canonical reduction systems of reducible braids, we define the outermost component of reducible braids, and then show that, for the reducible braids whose outermost component is simpler than the whole braid (including split braids), each element of its ultra summit set has a standard reduction system. This implies that, for such braids, finding a reduction system is as easy as finding a single element of the ultra summit set.