numberstoBn

NumberstoBn is a term used to describe a family of encodings that assign to each nonnegative integer k a corresponding element of the braid group B_n, for a fixed n ≥ 2. The aim is to provide a deterministic, compact representation of braids suitable for computational experiments in algebraic combinatorics and related fields. Because there is no single canonical encoding, different implementations define numberstoBn in different ways.

In a typical construction, one fixes an encoding rule that converts the base-2 (or other base) expansion of k into a word in the generators sigma_1, ..., sigma_{n-1} of B_n. For example, in a simple variant for B_3 one may map k to a braid word obtained by examining the binary digits of k and appending sigma_1 for certain positions and sigma_2 for others, or by concatenating a short fixed sequence of generators corresponding to each digit. The resulting product is then interpreted as an element of B_3 under the standard braid relations. Different rules produce different numberstoBn mappings.

Properties and uses: The map is typically neither injective nor surjective onto B_n; braid words are considered modulo relations, so many distinct integers may yield the same braid element. The construction is mainly used as a testbed for algorithms dealing with the word problem, normal forms, and length functions in braid groups, and for generating deterministic sequences of braids for benchmarking computational algebra software. Some researchers study how choices of encoding affect computational properties and efficiency.

See also: Braid group, B_n, braid word problem, computational group theory.