Invariants from Braid Theory

2008-12-16T16:12:41Z

VitrrAccna: zelvidompa

domlilatr
{{Manual TOC Sidebar}}

The ''braid length'' of a knot or a link <math>K</math> is the smallest number of crossings in a braid whose closure is <math>K</math>. <code>KnotTheory`</code> has some braid lengths preloaded:

{{Startup Note}}



{{HelpLine|
n = 1 |
in = <nowiki>BraidLength</nowiki> |
out= <nowiki>BraidLength[K] returns the braid length of the knot K, if known to KnotTheory`.</nowiki>}}


Note that the braid length of <math>K</math> is simply the length of the minimum braid representing <math>K</math> (see [[Braid Representatives]]):



{{InOut|
n = 2 |
in = <nowiki>K = Knot[9, 49]; {BraidLength[K], Crossings[BR[K]]}</nowiki> |
out= <nowiki>{11, 11}</nowiki>}}


{{Knot Image Pair|9_49|gif|10_136|gif}}

The ''braid index'' of a knot or a link <math>K</math> is the smallest number of strands in a braid whose closure is <math>K</math>. <code>KnotTheory`</code> has some braid indices preloaded:



{{HelpAndAbout|
n = 3 |
n1 = 4 |
in = <nowiki>BraidIndex</nowiki> |
out= <nowiki>BraidIndex[K] returns the braid index of the knot K, if known to KnotTheory`.</nowiki> |
about= <nowiki>The braid index data known to KnotTheory` is taken from Charles Livingston's "Table of Knot Invariants", http://www.indiana.edu/~knotinfo/.</nowiki>}}


Of the 250 knots with up to 10 crossings, only [[10_136]] has braid index smaller than the width of its minimum braid:



{{InOut|
n = 5 |
in = <nowiki>K = Knot[10, 136]; {BraidIndex[K], First@BR[K]}</nowiki> |
out= <nowiki>{4, 5}</nowiki>}}




{{Graphics|
n = 7 |
in = <nowiki>Show[BraidPlot[BR[K]]]</nowiki> |
img= Invariants_from_Braid_Theory_Out_6.gif |
out= <nowiki>-Graphics-</nowiki>}}

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Invariants from Braid Theory