Crossing matrices and Thurston's normal form for braids

Crossing Matrices and Thurston's Canonical Form for Braids

A Normal Form for Braids

We present a seemingly new normal form for braids, where every braid is expressed using a word in a regular language on some simple alphabet of elementary braids. This normal form stems from analysing the geometric action of braid groups on curves in a punctured disk.

Smith Normal Form and acyclic matrices

An approach, based on the Smith Normal Form, is introduced to study the spectra of symmetric matrices with a given graph. The approach serves well to explain how the path cover number (resp. diameter of a tree T ) is related to the maximal multiplicity MaxMult(T ) occurring for an eigenvalue of a symmetric matrix whose graph is T (resp. the minimal number q(T ) of distinct eigenvalues over the ...

Hex, Braids, the Crossing Rule, and XH-Search

We present XH-search, a Hex connection finding algorithm. XH-search extends Anshelevich’s H-search by incorporating a new crossing rule to find braids, connections built from overlapping subconnections.

Alternating Normal Forms for Braids and Locally Garside Monoids

We describe new types of normal forms for braid monoids, Artin– Tits monoids, and, more generally, all monoids in which divisibility has some convenient lattice properties (“locally Garside monoids”). We show that, in the case of braids, one of these normal forms turns out to coincide with the normal form introduced by Burckel and deduce that the latter can be computed easily. This approach lea...