andrewbartholomew62 / andrew-bartholomew-braid

A programme for calculating invariants and maipulating codes for various categories of knots and links

braid-programme

The braid programme was originally written to calculate matrix representations of virtual braid groups. It has evolved to include tasks that relate to virtual knots, long knots, welded knots and knotoids but that do not involve a braid representation. However, in the absence of anything better, the name 'braid' has been retained for the programme.

The tools provided by the programme support the following:

Alexander and Burau polynomial invariant
Alexander-like polynomial invariant of a classical doodle
Burau, or generalized Alexander, polynomial invariant
Commutative automorphism switch invariants
Finite-switch polynomial invariants (also known as rack polynomials)
Fixed-point invariant of the braid representation of a knot or doodle
Matrix-switch polynomial invariants
Quaternionic polynomial invariants
Sawollek's normalized Conway polynomial for a braid word
Weyl algebra switch polynomial invariants

Arrow polynomial of a classical or virtual knot, link, long knot, long virtual knot, knotoid or multi-knotoid
HOMFLY polynomial
Jones polynomial of a classical or virtual knot, link, knotoid or multi-knotoid
Kauffman's Affine Index Polynomial for virtual knots or knotoids
Kauffman bracket polynomial of a classical or virtual knot, link, knotoid or multi-knotoid
Turaev's extended bracket polynomial of a classical or virtual knotoid or multi-knotoid
Parity arrow polynomial of a classical or virtual knot or knotoid
Parity bracket polynomial of a classical or virtual knot or knotoid

Dynnikov test for the trivial braid
Hamiltonian circuits within classical or flat knot or link diagram
Vogel's algorithm for determining a braid word from a knot or link diagram

Dowker(-Thistlethwaite) code for a braid or labelled peer code
Gauss code for a labelled peer code or the closure of a braid word
Labelled immersion code for a braid word
Labelled peer code for
    braid word of a classical or virtual knot or link
    labelled immersion code of a classical or virtual knot or link
    Gauss code of a classical, virtual or flat link, knotoid, multi-knotoid or multi-linkoid
    Labelled peer code for the planar diagram description (PD code) of a classical or virtual link
    the n-parallel cable satellite of a knot's peer code