Braid Representatives

2009-05-22T07:33:53Z

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

{{Startup Note}}



{{HelpAndAbout|
n = 1 |
n1 = 2 |
in = <nowiki>BR</nowiki> |
out= <nowiki>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.</nowiki> |
about= <nowiki>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.</nowiki>}}




{{HelpLine|
n = 3 |
in = <nowiki>Mirror</nowiki> |
out= <nowiki>Mirror[br] return the mirror braid of br.</nowiki>}}


Thus for example,



{{In|
n = 4 |
in = <nowiki>br1 = BR[2, {-1, -1, -1}];</nowiki>}}




{{InOut|
n = 5 |
in = <nowiki>PD[br1]</nowiki> |
out= <nowiki>PD[X[6, 3, 1, 4], X[4, 1, 5, 2], X[2, 5, 3, 6]]</nowiki>}}




{{InOut|
n = 6 |
in = <nowiki>Jones[br1][q]</nowiki> |
out= <nowiki> -4 -3 1
-q + q + -
q</nowiki>}}




{{InOut|
n = 7 |
in = <nowiki>Mirror[br1]</nowiki> |
out= <nowiki>BR[2, {1, 1, 1}]</nowiki>}}


{{Knot Image Pair|T(5,4)|jpg|K11a362|gif}}

<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,



{{InOut|
n = 8 |
in = <nowiki>BR[TorusKnot[5, 4]]</nowiki> |
out= <nowiki>BR[4, {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}]</nowiki>}}




{{InOut|
n = 9 |
in = <nowiki>BR[Knot[11, Alternating, 362]]</nowiki> |
out= <nowiki>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}]</nowiki>}}


(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]]).

{{Knot Image Pair|10_1|gif|5_2|gif}}

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]]):



{{InOut|
n = 10 |
in = <nowiki>br2 = BR[Knot[10, 1]]</nowiki> |
out= <nowiki>BR[6, {-1, -1, -2, 1, -2, -3, 2, -3, -4, 3, 5, -4, 5}]</nowiki>}}




{{Graphics|
n = 12 |
in = <nowiki>Show[BraidPlot[CollapseBraid[br2]]]</nowiki> |
img= Braid_Representatives_Out_11.gif |
out= <nowiki>-Graphics-</nowiki>}}


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



{{Graphics|
n = 14 |
in = <nowiki>Show[BraidPlot[CollapseBraid[BR[Knot[5, 2]]]]]</nowiki> |
img= Braid_Representatives_Out_13.gif |
out= <nowiki>-Graphics-</nowiki>}}


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



{{InOut|
n = 15 |
in = <nowiki>pd = PD[Knot[5, 2]]</nowiki> |
out= <nowiki>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]]</nowiki>}}


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



{{Graphics|
n = 17 |
in = <nowiki>Show[BraidPlot[CollapseBraid[BR[pd]]]]</nowiki> |
img= Braid_Representatives_Out_16.gif |
out= <nowiki>-Graphics-</nowiki>}}


(Check [[Drawing Braids]] for information about the command <code>BraidPlot</code> and the related command <code>CollapseBraid</code>.)

{{note|Gittings}} T. A. Gittings, ''Minimum braids: a complete invariant of knots and links'', {{arXiv|math.GT/0401051}}.

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Braid Representatives