Braid Representatives: Difference between revisions

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{{Knot Image|K11a247|gif}}
⚫
<code>KnotTheory`</code> has the braid representatives of some knots and links pre-loaded. Thus for example,

⚫
<code>KnotTheory`</code> 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,


<!--$$BR[TorusKnot[5, 4]]$$-->
<!--$$BR[TorusKnot[5, 4]]$$-->
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out= <nowiki>BR[4, {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}]</nowiki>}}
out= <nowiki>BR[4, {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}]</nowiki>}}
<!--END-->
<!--END-->

<!--$$BR[Knot[11, Alternating, 247]]$$-->
<!--END-->

(As we see, Vogel's algorithm sometimes produces scary results. A <!--$Crossings[BR[Knot[11, Alternating, 247]]$--><!--END-->-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 {{ref|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]]):
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 {{ref|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]]):
Line 86: Line 93:
img= Braid_Representatives_Out_10.gif |
img= Braid_Representatives_Out_10.gif |
out= <nowiki>-Graphics-</nowiki>}}
out= <nowiki>-Graphics-</nowiki>}}
<!--END-->

{{Knot Image|5_2|gif}}

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

<!--$$Show[BraidPlot[CollapseBraid[BR[Knot[5, 2]]]]$$-->
<!--END-->

To force <code>KnotTheory`</code> to run Vogel's algorithm on [[5_2]], we first convert it to its <code>PD</code> form,

<!--$$pd = PD[Knot[5, 2]]$$-->
<!--END-->

and only then run <code>BR</code>:

<!--$$Show[BraidPlot[CollapseBraid[BR[pd]]]$$-->
<!--END-->
<!--END-->


Revision as of 14:59, 2 September 2005


KnotTheory`/Navigation 

The Mathematica Package KnotTheory`

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}]
K11a247.gif
K11a247

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}]


(As we see, Vogel's algorithm sometimes produces scary results. A -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[9]:= br2 = BR[Knot[10, 1]]
Out[9]= BR[6, {-1, -1, -2, 1, -2, -3, 2, -3, -4, 3, 5, -4, 5}]
In[11]:= Show[BraidPlot[CollapseBraid[br2]]]
Braid Representatives Out 10.gif
Out[11]= -Graphics-
5 2.gif
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


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


and only then run BR:


(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.

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