Symmetry And Group

Quilts: Central Extensions, Braid Actions and Finite Groups by Tim Hsu

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By Tim Hsu

Quilts are 2-complexes used to investigate activities and subgroups of the 3-string braid staff and related teams. This monograph establishes the basics of quilts and discusses connections with vital extensions, braid activities, and finite teams. such a lot effects haven't formerly seemed in a greatly to be had shape, and plenty of effects seem in print for the 1st time. This monograph is offered to graduate scholars, as quite a lot of historical past fabric is incorporated. The equipment and effects will be suitable to researchers drawn to endless teams, moonshine, significant extensions, triangle teams, dessins d'enfants, and monodromy activities of braid teams.

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4), so it remains to check that if A is a seam of Q, then (A,j)Vr ml = ( A , j ) Z p~ for r = 1,2 and all j (mod M). Fig. 5. Flows around a vertex (type 1) Now, consider a vertex of Q of type r (r = 1, 2) and collapsing index i that contains the seams A = A1, A 2 , . . , Ak in counterclockwise order, as shown in Fig. 5. -. + tk be the total flow into the vertex. 5) = ( A k , j + t l + . . + tk-1)Vr = ( A I , j + t). However, since the definition of collapsing index (Defn. 1) implies that ki = mr, and the definition of quilt diagram (Defn.

2 . 2 . A seam, m2 -- 2 We begin by describing the basic building block of our 2-complexes. Consider the double triangular region shown in Fig. 1. Such a region is called a seam. More formally, a seam is a 2-complex s constructed as follows. 1. s has four 0-cells, which are marked in Fig. 1 as dl and d2 (the dots of type 1 and 2), pt, and Pr. By convention, black dots are of type 1, and white dots are of type 2. 2. s has five 1-cells, which consist of one solid 1-cell, between dl and d2 (the midline of s), and four dotted 1-cells, between dl and Pt, dl and Pr, d2 and Pt, and dz and Pr (marked V1, Vx-1, V2-1, and V2, respectively).

We first define the case n -- c~ (Defn. 1), and later proceed to the general case (Defn. 6). 1. A (ml,m2,c~)-modular quilt Q is a set of seams with some of their dotted 1-cells identified (in the sense of a quotient space) according to the following rules. 1. For r = 1, 2, a dotted 1-cell marked Vr can only be identified with a dotted 1-cell marked V~-1. Furthermore, all identifications are cellular (n-cells with n-cells), and dots of type r (r = 1, 2) m a y only be identified with other dots of type r.

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