Arithmetic Operators and Functions for Elements

This section describes the basic arithmetic operations for elements of a braid group. Strictly speaking, all functions should be considered as functions on representatives of elements, that is, words in the Artin generators. The complexity of all operations is linear in the length of the words representing the input elements.

u * v : GrpBrdElt, GrpBrdElt -> GrpBrdElt

Given elements u and v belonging to the same braid group B, return the product uv as a new element of B.

u *:= v : GrpBrdElt, GrpBrdElt ->

Given elements u and v belonging to the same braid group B, replace u with the product uv.

u / v : GrpBrdElt, GrpBrdElt -> GrpBrdElt

Given elements u and v belonging to the same braid group B, return the product uv^(-1) as a new element of B.

u /:= v : GrpBrdElt, GrpBrdElt ->

Given elements u and v belonging to the same braid group B, replace u with the product uv^(-1).

u ^ n : GrpBrdElt, RngIntElt -> GrpBrdElt

Given an element u of a braid group B and an integer n, return the power u^n as a new element of B.

u ^:= n : GrpBrdElt, RngIntElt ->

Given an element u of a braid group B and an integer n, replace u with the power u^n.

u ^ v : GrpBrdElt, GrpBrdElt -> GrpBrdElt

Given elements u and v belonging to the same braid group B, return the conjugate u^v := v^(-1)uv as a new element of B.

u ^:= v : GrpBrdElt, GrpBrdElt ->

Given elements u and v belonging to the same braid group B, replace u with the conjugate u^v := v^(-1)uv.

Inverse(u) : GrpBrdElt -> GrpBrdElt

Given an element u of a braid group B, return its inverse u^(-1) as a new element of B.

Inverse(~u) : GrpBrdElt ->

Given an element u of a braid group B, replace u with its inverse u^(-1).

LeftConjugate(u, v) : GrpBrdElt, GrpBrdElt -> GrpBrdElt

Given elements u and v belonging to the same braid group B, return the "left conjugate" vuv^(-1) as a new element of B.

LeftConjugate(~u, v) : GrpBrdElt, GrpBrdElt -> GrpBrdElt

Given elements u and v belonging to the same braid group B, replace u with the "left conjugate" vuv^(-1).

LeftDiv(u, v) : GrpBrdElt, GrpBrdElt -> GrpBrdElt

Given elements u and v belonging to the same braid group B, return the product u^(-1)v as a new element of B.

LeftDiv(u, ~v) : GrpBrdElt, GrpBrdElt -> GrpBrdElt

Given elements u and v belonging to the same braid group B, replace v with the product u^(-1)v.