Lightly Documented Features: Difference between revisions

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<!--$$?NumberOfKnots$$-->
<!--$$?NumberOfKnots$$-->
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{{HelpLine|
{{Help1|n=1|s=NumberOfKnots}}
n = 2 |
⚫
NumberOfKnots[type] return the number of knots of a given type.
in = <nowiki>NumberOfKnots</nowiki> |
{{Help2}}
⚫
out= <nowiki>NumberOfKnots[n] returns the number of knots with n crossings.
NumberOfKnots[n, Alternating&#124;NonAlternating] returns the number of knots of the specified type.</nowiki>}}
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<!--$$NumberOfKnots[16, NonAlternating]$$-->
<!--$$NumberOfKnots[16, NonAlternating]$$-->
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{{InOut1|n=2}}
{{InOut|
n = 3 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>NumberOfKnots[16, NonAlternating]</nowiki></pre>
in = <nowiki>NumberOfKnots[16, NonAlternating]</nowiki> |
{{InOut2|n=2}}<pre style="border: 0px; padding: 0em"><nowiki>1008906</nowiki></pre>
out= <nowiki>1008906</nowiki>}}
{{InOut3}}
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<!--$$?MorseLink$$-->
<!--$$?AlternatingQ$$-->
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{{HelpLine|
{{HelpAndAbout1|n=3|s=MorseLink}}
n = 4 |
MorseLink[K] returns a presentation of the oriented link K, composed, in successive order, of the following 'events': Cup[m,n] is a directed creation, starting at strand position n, towards position m, where m and n differ by 1. X[n,a = {Over/Under}, b = {Up/Down}, c={Up/Down}] is a crossing with lower-left edge at strand n, a determines whether the strand running bottom-left to top-right is over/under the crossing, b and c give the directions of the bottom-left and bottom-right strands respectively through the crossing. Cap[m,n] is a directed cap, from strand m to strand n.
in = <nowiki>AlternatingQ</nowiki> |
{{HelpAndAbout2|n=4|s=MorseLink}}
out= <nowiki>AlternatingQ[D] returns True iff the knot/link diagram D is alternating.</nowiki>}}
MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005.
{{HelpAndAbout3}}
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Among the knots with up to 11 crossings, 564 are alternating and 238 are not:
<!--$$MorseLink[Knot[3, 1]]$$-->
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{{InOut1|n=5}}
<pre style="color: red; border: 0px; padding: 0em"><nowiki>MorseLink[Knot[3, 1]]</nowiki></pre>
{{InOut2|n=5}}<pre style="border: 0px; padding: 0em"><nowiki>MorseLink[1 ? 2, 4 ? 3, X[2, Under, Up, Up], X[2, Under, Up, Up],
X[2, Under, Up, Up], 2 ? 1, 1 ? 2]</nowiki></pre>
{{InOut3}}
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<!--$$?DrawMorseLink$$-->
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{{HelpAndAbout1|n=6|s=DrawMorseLink}}
DrawMorseLink[L] returns a drawing of the knot or link L as a "Morse Link". For diagrams with a large number of crossings, it may be helpful to use one or both of the options as in DrawMorseLink[L, Gap -> g, ArrowSize -> as ], with 0 < as, g < 1, where g controls the amount of white space at each crossing, and as controls the size of the orientation arrows.
{{HelpAndAbout2|n=7|s=DrawMorseLink}}
DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
{{HelpAndAbout3}}
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<!--$$Show[DrawMorseLink[Link[11, Alternating, 548]]]$$-->
<!--$$Total[AlternatingQ /@ AllKnots[{0,11}]]$$-->
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{{InOut|
{{Graphics1|n=8}}
n = 5 |
<pre style="color: red; border: 0px; padding: 0em"><nowiki>Show[DrawMorseLink[Link[11, Alternating, 548]]]</nowiki></pre>
in = <nowiki>Total[AlternatingQ /@ AllKnots[{0,11}]]</nowiki> |
{{Graphics2|n=8|imagename=Lightly_Documented_Features_Out_8.gif}}
out= <nowiki>238 False + 564 True</nowiki>}}
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Latest revision as of 17:24, 21 February 2013


KnotTheory`/Navigation 

The Mathematica Package KnotTheory`

(For In[1] see Setup)

In[2]:= ?NumberOfKnots
NumberOfKnots[n] returns the number of knots with n crossings. NumberOfKnots[n, Alternating|NonAlternating] returns the number of knots of the specified type.
In[3]:= NumberOfKnots[16, NonAlternating]
Out[3]= 1008906
In[4]:= ?AlternatingQ
AlternatingQ[D] returns True iff the knot/link diagram D is alternating.

Among the knots with up to 11 crossings, 564 are alternating and 238 are not:

In[5]:= Total[AlternatingQ /@ AllKnots[{0,11}]]
Out[5]= 238 False + 564 True
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