Invariants from Braid Theory: Difference between revisions
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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: |
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: |
Revision as of 18:30, 28 August 2007
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[2]:=
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K = Knot[9, 49]; {BraidLength[K], Crossings[BR[K]]}
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Out[2]=
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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[5]:=
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K = Knot[10, 136]; {BraidIndex[K], First@BR[K]}
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Out[5]=
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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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