Braid Representatives

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

(For In[1] see Setup)

In[1]:= ?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[2]:= 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[3]:= ?Mirror Mirror[br] return the mirror braid of br.

Thus for example,

In[4]:=

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

In[5]:=	PD[br1]
Out[5]=	PD[X[6, 3, 1, 4], X[4, 1, 5, 2], X[2, 5, 3, 6]]

In[6]:=	Jones[br1][q]
Out[6]=	-4 -3 1 -q + q + - q

In[7]:=	Mirror[br1]
Out[7]=	BR[2, {1, 1, 1}]

K11a362

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,

In[8]:=	BR[TorusKnot[5, 4]]
Out[8]=	BR[4, {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}]

In[9]:=	BR[Knot[11, Alternating, 362]]
Out[9]=	BR[10, {1, 2, -3, -4, 5, 6, 5, 4, 3, -2, -1, -4, 3, -2, -4, 3, 5, 4, -6, 7, -6, 5, 8, 7, 6, 5, -4, -3, 2, 5, -6, 9, -8, 7, -6, 5, 4, -3, 5, 6, 5, 4, 5, -7, 8, -7, 6, 5, -9, -8, -7}]

(As we see, Vogel's algorithm sometimes produces scary results. A $Failed-crossings braid representative for an 11-crossing knot, in this case).

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):

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[12]:=	Show[BraidPlot[CollapseBraid[br2]]]
Out[12]=	-Graphics-

5_2

Already for the knot 5_2 the minimum braid is shorter than the braid produced by Vogel's algorithm. Indeed, the minimum braid is

In[13]:=	Show[BraidPlot[CollapseBraid[BR[Knot[5, 2]]]]
Out[13]=	$Failed

To force KnotTheory` to run Vogel's algorithm on 5_2, we first convert it to its PD form,

In[14]:=	pd = PD[Knot[5, 2]]
Out[14]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 10, 6, 1], X[9, 6, 10, 7], X[7, 2, 8, 3]]

and only then run BR:

In[15]:=	Show[BraidPlot[CollapseBraid[BR[pd]]]
Out[15]=	$Failed

(Check Drawing Braids for information about the command BraidPlot and the related command CollapseBraid.)

[Gittings] ^  T. A. Gittings, Minimum braids: a complete invariant of knots and links, arXiv:math.GT/0401051.