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4.4 Braid Representatives

Every knot and every link is the closure of a braid. KnotTheory` can also represent knots and links as braid closures:

In[1]

In[2]:= ?BR
BR stands for Braid Representative. BR[k,l] represents a braid on k strands with crossings l={i1,i2,...}, where a positive index i within the list l indicates a right-handed crossing between strand number i and strand number i+1 and a negative i indicates a left handed crossing between strands numbers |i| and |i|+1. Each ij can also be a list of non-adjacent (i.e., commuting) indices. BR also acts as a "type caster": BR[K] will return a braid whose closure is K if K is given in any format that KnotTheory` understands. BR[K] where K is is a named knot with up to 10 crossings returns a minimum braid representative for that knot.

In[3]:= BR::about
The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See his article on the subject at arXiv:math.GT/0401051. Vogel's algorithm was implemented by Dan Carney in the summer of 2005 at the University of Toronto.

In[4]:= ?Mirror
Mirror[br] return the mirror braid of br.

Thus for example,

In[5]:=

br1 = BR[2, {-1, -1, -1}];

In[6]:=	PD[br1, q]
Out[6]=	PD[BR[2, {-1, -1, -1}], q]

In[7]:=	Jones[br1][q]
Out[7]=	-4 -3 1 -q + q + - q

In[8]:=	Mirror[br1]
Out[8]=	BR[2, {1, 1, 1}]

KnotTheory` has the braid representatives of some knots and links pre-loaded. Thus for example,

In[9]:=	BR[TorusKnot[5, 4]]
Out[9]=	BR[4, {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}]

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' [Gi]). 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 (number of strands, also see Section 7.1) 6:

In[10]:=	br2 = BR[Knot[10, 1]]
Out[10]=	BR[6, {-1, -1, -2, 1, -2, -3, 2, -3, -4, 3, 5, -4, 5}]

In[11]:=	Show[BraidPlot[CollapseBraid[br2]]]
Out[11]=	-Graphics-

(Check Section 5.2 for information about the command BraidPlot and the related command CollapseBraid.)