Design and serial construction of digraph braids

Abstract

Braiding is a traditional art used to enhance both the decorative and structural properties of any sort of stranded material, and braids are designed with attention to aesthetic principles, structural cohesion, and ease of construction. This work will determine, for a family of braids which are determined by directed graph structures, how these desiderata can be associated with a mathematical property of the underlying directed graphs. Of particular interest in this investigation is the notion of serial constructability, in which individual strands are laid down separately, and the closely related property of fault tolerance. In addition, the established property of braid decomposability is explored through the lens of this relationship between braid cohesiveness and digraph properties. These results are demonstrated for braids with a prescribed crossing sequence, and the potential extensions to braids with arbitrary sequences of crossings are explored.

Since a braid is described by both the sequence of permutations performed on the strands and the choice for each crossing of which strand lies on top, this study is chiefly concerned with the modification of the strand-precedence rules associated with a single permutation sequence to produce a family of distinct braids. The permutation sequence investigated will be that which is used in traditional three-strand braids and generalizations thereof, while the precedence rules on strands will be dictated by the associated directed graph.

Keywords:

• braids
• digraphs
• knot theory
• tournaments
• acyclic digraphs
• decomposability

AMS Subject Classifications:

• 05C20
• 00A66
• 57M25
• 20F36

Acknowledgements

This work was extensively improved by suggestions from the editors and reviewers and by information provided by braidcrafter Jacqui Carey. Thanks are also due to the organizers and referees of Bridges 2012: Mathematics, Music, Art, Architecture, Culture for their suggestions, and for providing a venue for the presentation of an abbreviated version of this work.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. The term transitive is often used to describe tournaments without cycles; since the similarly named but unrelated property of vertex-transitivity is used in this work, such tournaments shall be referred to as acyclic to reduce confusion.

2. The indegree of a vertex in a directed graph is the number of edges which are directed into it; likewise, the outdegree is the number of outbound edges.