Invariants from Braid Theory: Difference between revisions
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The |
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: |
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{{Startup Note}} |
{{Startup Note}} |
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<!--$$?BraidLength$$--> |
<!--$$?BraidLength$$--> |
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n = 2 | |
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in = <nowiki>BraidLength</nowiki> | |
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out= <nowiki>BraidLength[K] returns the braid length of the knot K, if known to KnotTheory`.</nowiki>}} |
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<!--$$K = Knot[9, 49]; {BraidLength[K], Crossings[BR[K]]}$$--> |
<!--$$K = Knot[9, 49]; {BraidLength[K], Crossings[BR[K]]}$$--> |
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{{InOut| |
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n = 3 | |
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in = <nowiki>K = Knot[9, 49]; {BraidLength[K], Crossings[BR[K]]}</nowiki> | |
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out= <nowiki>{11, 11}</nowiki>}} |
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{{Knot Image Pair|9_49|gif|10_136|gif}} |
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<!--$$?BraidIndex$$--> |
<!--$$?BraidIndex$$--> |
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{{HelpAndAbout| |
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n = 4 | |
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n1 = 5 | |
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in = <nowiki>BraidIndex</nowiki> | |
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out= <nowiki>BraidIndex[K] returns the braid index of the knot K, if known to KnotTheory`.</nowiki> | |
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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>}} |
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<!--$$K = Knot[10, 136]; {BraidIndex[K], First@BR[K]}$$--> |
<!--$$K = Knot[10, 136]; {BraidIndex[K], First@BR[K]}$$--> |
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{{InOut| |
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n = 6 | |
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in = <nowiki>K = Knot[10, 136]; {BraidIndex[K], First@BR[K]}</nowiki> | |
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out= <nowiki>{4, 5}</nowiki>}} |
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<!--$$Show[BraidPlot[BR[K]]]$$--> |
<!--$$Show[BraidPlot[BR[K]]]$$--> |
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{{Graphics| |
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n = 7 | |
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in = <nowiki>Show[BraidPlot[BR[K]]]</nowiki> | |
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img= Invariants_from_Braid_Theory_Out_7.gif | |
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out= <nowiki>-Graphics-</nowiki>}} |
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Latest revision as of 18:20, 21 February 2013
The braid length of a knot or a link is the smallest number of crossings in a braid whose closure is . KnotTheory` has some braid lengths preloaded:
(For In[1] see Setup)
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Note that the braid length of is simply the length of the minimum braid representing (see Braid Representatives):
In[3]:=
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K = Knot[9, 49]; {BraidLength[K], Crossings[BR[K]]}
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Out[3]=
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{11, 11}
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 9_49 |
 10_136 |
The braid index of a knot or a link is the smallest number of strands in a braid whose closure is . KnotTheory` has some braid indices preloaded:
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Of the 250 knots with up to 10 crossings, only 10_136 has braid index smaller than the width of its minimum braid:
In[6]:=
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K = Knot[10, 136]; {BraidIndex[K], First@BR[K]}
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Out[6]=
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{4, 5}
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In[7]:=
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Show[BraidPlot[BR[K]]]
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Out[7]=
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-Graphics-
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