Lightly Documented Features: Difference between revisions

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{{HelpLine|
{{HelpLine|
n = 1 |
n = 2 |
in = <nowiki>NumberOfKnots</nowiki> |
in = <nowiki>NumberOfKnots</nowiki> |
out= <nowiki>NumberOfKnots[type] return the number of knots of a given type.</nowiki>}}
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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{{InOut|
{{InOut|
n = 2 |
n = 3 |
in = <nowiki>NumberOfKnots[16, NonAlternating]</nowiki> |
in = <nowiki>NumberOfKnots[16, NonAlternating]</nowiki> |
out= <nowiki>1008906</nowiki>}}
out= <nowiki>1008906</nowiki>}}
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<!--$$?MorseLink$$-->
<!--$$?AlternatingQ$$-->
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{{HelpLine|
{{HelpAndAbout|
n = 3 |
n = 4 |
⚫
in = <nowiki>AlternatingQ</nowiki> |
n1 = 4 |
out= <nowiki>AlternatingQ[D] returns True iff the knot/link diagram D is alternating.</nowiki>}}
⚫
in = <nowiki>MorseLink</nowiki> |
out= <nowiki>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.</nowiki> |
about= <nowiki>MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005.</nowiki>}}
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Among the knots with up to 11 crossings, 564 are alternating and 238 are not:
<!--$$MorseLink[Knot[3, 1]]$$-->

<!--$$Total[AlternatingQ /@ AllKnots[{0,11}]]$$-->
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{{InOut|
{{InOut|
n = 5 |
n = 5 |
in = <nowiki>MorseLink[Knot[3, 1]]</nowiki> |
in = <nowiki>Total[AlternatingQ /@ AllKnots[{0,11}]]</nowiki> |
⚫
out= <nowiki>238 False + 564 True</nowiki>}}
out= <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>}}
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<!--$$?DrawMorseLink$$-->
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{{HelpAndAbout|
n = 6 |
n1 = 7 |
in = <nowiki>DrawMorseLink</nowiki> |
out= <nowiki>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.</nowiki> |
about= <nowiki>DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.</nowiki>}}
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<!--$$Show[DrawMorseLink[Link[11, Alternating, 548]]]$$-->
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{{Graphics|
n = 9 |
in = <nowiki>Show[DrawMorseLink[Link[11, Alternating, 548]]]</nowiki> |
img= Lightly_Documented_Features_Out_8.gif |
⚫
out= <nowiki>-Graphics-</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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