Invariants from Braid Theory: Difference between revisions
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{{Help1|n=2|s=BraidLength}} |
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BraidLength[K] returns the braid length of the knot K, if known to KnotTheory`. |
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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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{{InOut1|n=3}} |
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K = Knot[9, 49]; {BraidLength[K], Crossings[BR[K]]} |
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{{InOut2|n=3}}<pre style="border: 0px; padding: 0em"><nowiki>{11, 11}</nowiki></pre> |
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<!--$$?BraidIndex$$--> |
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{{HelpAndAbout1|n=4|s=BraidIndex}} |
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BraidIndex[K] returns the braid index of the knot K, if known to KnotTheory`. |
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{{HelpAndAbout2|n=5|s=BraidIndex}} |
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The braid index data known to KnotTheory` is taken from Charles Livingston's "Table of Knot Invariants", http://www.indiana.edu/~knotinfo/. |
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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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{{InOut1|n=6}} |
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K = Knot[10, 136]; {BraidIndex[K], First@BR[K]} |
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{{InOut2|n=6}}<pre style="border: 0px; padding: 0em"><nowiki>{4, 5}</nowiki></pre> |
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<!--$$Show[BraidPlot[BR[K]]]$$--> |
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{{Graphics1|n=7}} |
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Show[BraidPlot[BR[K]]] |
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{{Graphics2|n=7|imagename=Invariants_from_Braid_Theory_Out_7.gif}} |
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Revision as of 22:02, 24 August 2005
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)
In[2]:= ?BraidLength
BraidLength[K] returns the braid length of the knot K, if known to KnotTheory`. |
Note that the braid length of is simply the length of the minimum braid representing (see Braid Representatives):
| In[3]:= |
K = Knot[9, 49]; {BraidLength[K], Crossings[BR[K]]} |
| Out[3]= | {11, 11}
|
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:
In[4]:= ?BraidIndex
BraidIndex[K] returns the braid index of the knot K, if known to KnotTheory`. |
In[5]:= BraidIndex::about
The braid index data known to KnotTheory` is taken from Charles Livingston's "Table of Knot Invariants", http://www.indiana.edu/~knotinfo/. |
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]:= |
K = Knot[10, 136]; {BraidIndex[K], First@BR[K]} |
| Out[6]= | {4, 5}
|
| In[7]:= |
Show[BraidPlot[BR[K]]] |
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| Out[7]= | -Graphics- |
