Invariants from Braid Theory: Difference between revisions
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{{Help1|n= |
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BraidLength[K] returns the braid length of the knot K, if known to KnotTheory`. |
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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K = Knot[9, 49]; {BraidLength[K], Crossings[BR[K]]} |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>K = Knot[9, 49]; {BraidLength[K], Crossings[BR[K]]}</nowiki></pre> |
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{{InOut2|n=2}}<pre style="border: 0px; padding: 0em"><nowiki>{11, 11}</nowiki></pre> |
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<!--$$?BraidIndex$$--> |
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BraidIndex[K] returns the braid index of the knot K, if known to KnotTheory`. |
BraidIndex[K] returns the braid index of the knot K, if known to KnotTheory`. |
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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/. |
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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K = Knot[10, 136]; {BraidIndex[K], First@BR[K]} |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>K = Knot[10, 136]; {BraidIndex[K], First@BR[K]}</nowiki></pre> |
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{{InOut2|n=5}}<pre style="border: 0px; padding: 0em"><nowiki>{4, 5}</nowiki></pre> |
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{{InOut3}} |
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<!--$$Show[BraidPlot[BR[K]]]$$--> |
<!--$$Show[BraidPlot[BR[K]]]$$--> |
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Show[BraidPlot[BR[K]]] |
Show[BraidPlot[BR[K]]] |
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{{Graphics2|n=6|imagename=Invariants_from_Braid_Theory_Out_6.gif}} |
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Revision as of 20:43, 27 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[1]:= ?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[2]:= |
K = Knot[9, 49]; {BraidLength[K], Crossings[BR[K]]}
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| Out[2]= | {11, 11}
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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[3]:= ?BraidIndex
BraidIndex[K] returns the braid index of the knot K, if known to KnotTheory`. |
In[4]:= 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[5]:= |
K = Knot[10, 136]; {BraidIndex[K], First@BR[K]}
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| Out[5]= | {4, 5}
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| In[6]:= |
Show[BraidPlot[BR[K]]] |
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| Out[6]= | -Graphics- |
