Arc Presentations: Difference between revisions

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<!--$$ap = ArcPresentation["K11n11"]$$-->
<!--$$ap = ArcPresentation["K11n11"]$$-->
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{{InOut|
n = 2 |
in = <nowiki>ap = ArcPresentation["K11n11"]</nowiki> |
out= <nowiki>ArcPresentation[{12, 2}, {1, 10}, {3, 9}, {5, 11}, {9, 12}, {4, 8},
{2, 5}, {11, 7}, {8, 6}, {7, 4}, {10, 3}, {6, 1}]</nowiki>}}
<!--END-->
<!--END-->


<!--$$Draw[ap]$$-->
<!--$$Draw[ap]$$-->
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{{Graphics|
n = 4 |
in = <nowiki>Draw[ap]</nowiki> |
img= Arc_Presentations_Out_3.gif |
out= <nowiki>-Graphics-</nowiki>}}
<!--END-->
<!--END-->


<!--$$ap0 = ArcPresentation["K11n11", Reduce -> 0]$$-->
<!--$$ap0 = ArcPresentation["K11n11", Reduce -> 0]$$-->
<!--Robot Land, no human edits to "END"-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 5 |
in = <nowiki>ap0 = ArcPresentation["K11n11", Reduce -> 0]</nowiki> |
out= <nowiki>ArcPresentation[{13, 19}, {20, 23}, {19, 22}, {15, 14}, {14, 2},
{1, 13}, {3, 12}, {2, 4}, {16, 18}, {17, 15}, {8, 16}, {12, 17},
{5, 7}, {4, 6}, {7, 11}, {6, 8}, {18, 10}, {11, 9}, {10, 21},
{9, 20}, {21, 5}, {22, 3}, {23, 1}]</nowiki>}}
<!--END-->
<!--END-->


<!--$$?Draw$$-->
<!--$$?Draw$$-->
<!--Robot Land, no human edits to "END"-->
<!--Robot Land, no human edits to "END"-->
{{HelpLine|
n = 6 |
in = <nowiki>Draw</nowiki> |
out= <nowiki>Draw[ap] draws the Arc Presentation ap. Draw[ap, OverlayMatrix -> M] overlays the matrix M on top of that draw.</nowiki>}}
<!--END-->
<!--END-->


<!--$$Draw[ap0]$$-->
<!--$$Draw[ap0]$$-->
<!--Robot Land, no human edits to "END"-->
<!--Robot Land, no human edits to "END"-->
{{Graphics|
n = 8 |
in = <nowiki>Draw[ap0]</nowiki> |
img= Arc_Presentations_Out_7.gif |
out= <nowiki>-Graphics-</nowiki>}}
<!--END-->
<!--END-->


Line 34: Line 64:
)$$-->
)$$-->
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{{In|
n = 9 |
in = <nowiki>Reflect[ap_ArcPresentation] := ArcPresentation @@ (
(Last /@ Sort[Reverse /@ Position[ap, #]]) & /@ Range[Length[ap]]
)</nowiki>}}
<!--END-->
<!--END-->


<!--$$Reflect[ap] // Draw$$-->
<!--$$Reflect[ap] // Draw$$-->
<!--Robot Land, no human edits to "END"-->
<!--Robot Land, no human edits to "END"-->
{{Graphics|
n = 11 |
in = <nowiki>Reflect[ap] // Draw</nowiki> |
img= Arc_Presentations_Out_10.gif |
out= <nowiki>-Graphics-</nowiki>}}
<!--END-->
<!--END-->


Line 57: Line 97:
$$-->
$$-->
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{{In|
n = 12 |
in = <nowiki>MinesweeperMatrix[ap_ArcPresentation] := Module[
{l, CurrentRow, c1, c2, k, s},
l = Length[ap];
CurrentRow = Table[0, {l}];
Table[
{c1, c2} = Sort[ap[[k]]];
s = Sign[{-1, 1}.ap[[k]]];
Do[
CurrentRow[[c]] += s,
{c, c1, c2 - 1}
];
CurrentRow,
{k, l}
]
];
</nowiki>}}
<!--END-->
<!--END-->


<!--$$Draw[ap, OverlayMatrix -> MinesweeperMatrix[ap]]$$-->
<!--$$Draw[ap, OverlayMatrix -> MinesweeperMatrix[ap]]$$-->
<!--Robot Land, no human edits to "END"-->
<!--Robot Land, no human edits to "END"-->
{{Graphics|
n = 14 |
in = <nowiki>Draw[ap, OverlayMatrix -> MinesweeperMatrix[ap]]</nowiki> |
img= Arc_Presentations_Out_13.gif |
out= <nowiki>-Graphics-</nowiki>}}
<!--END-->
<!--END-->


<!--$${Det[t^MinesweeperMatrix[ap]], Alexander[ap][t]} // Factor$$-->
<!--$${Det[t^MinesweeperMatrix[ap]], Alexander[ap][t]} // Factor$$-->
<!--Robot Land, no human edits to "END"-->
<!--Robot Land, no human edits to "END"-->
{{InOut|
n = 15 |
in = <nowiki>{Det[t^MinesweeperMatrix[ap]], Alexander[ap][t]} // Factor</nowiki> |
out= <nowiki> 11 2 2 3 4 5 6
{(-1 + t) t (1 - 5 t + 13 t - 17 t + 13 t - 5 t + t ),
2 3 4 5 6
1 - 5 t + 13 t - 17 t + 13 t - 5 t + t
-------------------------------------------}
3
t</nowiki>}}
<!--END-->
<!--END-->

Revision as of 22:03, 1 December 2007


KnotTheory`/Navigation 

The Mathematica Package KnotTheory`

(For In[1] see Setup)

In[1]:= ?ArcPresentation
ArcPresentation[{a1,b1}, {a2, b2}, ..., {an,bn}] is an arc presentation of a knot (as often used in the realm of Heegaard-Floer homologies), where ai is horizontal arc at row i connects column ai to column bi. ArcPresentation[K] returns an arc presentation of the knot K. ArcPresentation[K, Reduce -> r] attemps at most r reduction steps (using a naive reduction algorithm) following a naive creation of some arc presentation for K.
In[2]:= ap = ArcPresentation["K11n11"]
Out[2]= ArcPresentation[{12, 2}, {1, 10}, {3, 9}, {5, 11}, {9, 12}, {4, 8}, {2, 5}, {11, 7}, {8, 6}, {7, 4}, {10, 3}, {6, 1}]
In[4]:= Draw[ap]
Arc Presentations Out 3.gif
Out[4]= -Graphics-
In[5]:= ap0 = ArcPresentation["K11n11", Reduce -> 0]
Out[5]= ArcPresentation[{13, 19}, {20, 23}, {19, 22}, {15, 14}, {14, 2}, {1, 13}, {3, 12}, {2, 4}, {16, 18}, {17, 15}, {8, 16}, {12, 17}, {5, 7}, {4, 6}, {7, 11}, {6, 8}, {18, 10}, {11, 9}, {10, 21}, {9, 20}, {21, 5}, {22, 3}, {23, 1}]
In[6]:= ?Draw
Draw[ap] draws the Arc Presentation ap. Draw[ap, OverlayMatrix -> M] overlays the matrix M on top of that draw.
In[8]:= Draw[ap0]
Arc Presentations Out 7.gif
Out[8]= -Graphics-
In[9]:= Reflect[ap_ArcPresentation] := ArcPresentation @@ ( (Last /@ Sort[Reverse /@ Position[ap, #]]) & /@ Range[Length[ap]] )
In[11]:= Reflect[ap] // Draw
Arc Presentations Out 10.gif
Out[11]= -Graphics-
In[12]:= MinesweeperMatrix[ap_ArcPresentation] := Module[ {l, CurrentRow, c1, c2, k, s}, l = Length[ap]; CurrentRow = Table[0, {l}]; Table[ {c1, c2} = Sort[ap[[k]]]; s = Sign[{-1, 1}.ap[[k]]]; Do[ CurrentRow[[c]] += s, {c, c1, c2 - 1} ]; CurrentRow, {k, l} ] ];
In[14]:= Draw[ap, OverlayMatrix -> MinesweeperMatrix[ap]]
Arc Presentations Out 13.gif
Out[14]= -Graphics-
In[15]:= {Det[t^MinesweeperMatrix[ap]], Alexander[ap][t]} // Factor
Out[15]= 11 2 2 3 4 5 6 {(-1 + t) t (1 - 5 t + 13 t - 17 t + 13 t - 5 t + t ), 2 3 4 5 6 1 - 5 t + 13 t - 17 t + 13 t - 5 t + t -------------------------------------------} 3 t
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