From Knot Atlas
Every knot and every link is the closure of a braid.
KnotTheory` can also represent knots and links as braid closures:
(For In see Setup)
Thus for example,
KnotTheory` has the braid representatives of some knots and links pre-loaded, and for all other knots and links it will find a braid representative using Vogel's algorithm. Thus for example,
(As we see, Vogel's algorithm sometimes produces scary results. A 51-crossings braid representative for an 11-crossing knot, in the case of K11a362).
The minimum braid representative of a given knot is a braid representative for that knot which has a minimal number of braid crossings and within those braid representatives with a minimal number of braid crossings, it has a minimal number of strands (full details are in [Gittings]). Thomas Gittings kindly provided us the minimum braid representatives for all knots with up to 10 crossings. Thus for example, the minimum braid representative for the knot 10_1 has length (number of crossings) 13 and width 6 (number of strands, also see Invariants from Braid Theory):
Already for the knot 5_2 the minimum braid is shorter than the braid produced by Vogel's algorithm. Indeed, the minimum braid is
KnotTheory` to run Vogel's algorithm on 5_2, we first convert it to its
and only then run
(Check Drawing Braids for information about the command
BraidPlot and the related command