VasCalc Results - ChordMod4T: Difference between revisions

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This page documents some computational results obtained by using the <b>ChordsMod4T</b> software component of the [[VasCalc]] project. We denote by <math>{\mathcal A}(l,m,n)</math> the formal (rational) vector space generated by all chord diagrams with <math>n</math> chords on a skeleton of <math>l</math> lines and <math>m</math> circles, modulo the <math>4T</math> relations. The algorithm is straightforward: first all such diagrams are constructed, then relations are introduced iteratively over all possible terms in a <math>4T</math> relation. See the [[VasCalc Documentation - ChordsMod4T| documentation]] for further information about the program and how these results were generated.
This page documents some computational results obtained by using the <b>ChordsMod4T</b> software component of the [[VasCalc]] project. We denote by <math>{\mathcal A}_n(l,m)</math> the formal (rational) vector space generated by all chord diagrams with <math>n</math> chords on a skeleton of <math>l</math> lines and <math>m</math> circles, modulo the <math>4T</math> relations. The algorithm is straightforward: first all such diagrams are constructed, then relations are introduced iteratively over all possible terms in a <math>4T</math> relation. See the [[VasCalc Documentation - ChordsMod4T| documentation]] for further information about the program and how these results were generated.


===Some Dimensions of <math>{\mathcal A}(l,m,n)</math>===
===Some Dimensions of <math>{\mathcal A}_n(l,m)</math>===


The tables below give the dimensions of <math>{\mathcal A}(l,m,n)</math> for various values of <math>l</math>, <math>m</math> and <math>n</math>. Each table corresponds to a fixed value of <math>n</math> - that is, the number of chords.
The tables below give the dimensions of <math>{\mathcal A}_n(l,m)</math> for various values of <math>l</math>, <math>m</math> and <math>n</math>. Each table corresponds to a fixed value of <math>n</math> - that is, the number of chords.


'''This is work in progress - these numbers are not reliable yet!!!'''


{| border="1"
{| border="1" align=center cellspacing=0
|+ Dimensions of chord diagram spaces with 2 chords
|+ Dimensions of chord diagram spaces with 2 chords
!rowspan="2" width="100px"|Number of Lines
!rowspan="2" width="100px"|Number of Lines
Line 13: Line 14:
||0||1||2||3||4||5||6||7||8||9
||0||1||2||3||4||5||6||7||8||9
|- align=right
|- align=right
|0||0||2||8||27||69||145||272||469||758||1164
|0||0|| 2||8||24||59||125||237||413||674||1044
|- align=right
|- align=right
|1||2||8||26||68||145||272||469||758||1164||1715
|1||2||8||24||59||125||237||413||674||1044||1550
|- align=right
|- align=right
|2||9||25||66||144||272||469||758||1164||1715||2442
|2||9||25||60||126||238||414||675||1045||1551||2223
|- align=right
|- align=right
|3||28||63||141||271||469||758||1164||1715||2442||3379
|3||28||63||129||241||417||678||1048||1554||2226||3097
|- align=right
|- align=right
|4||69||135||267||468||758||1164||1715||2442||3379||4563
|4||69||135||247||423||684||1054||1560||2232||3103||4209
|- align=right
|- align=right
|5||145||257||463||757||1164||1715||2442||3379||4563||6034
|5||145||257||433||694||1064||1570||2242||3113||4219||5599
|- align=right
|- align=right
|6||272||448||751||1163||1715||2442||3379||4563||6034||7835
|6||272||448||709||1079||1585||2257||3128||4234||5614||7310
|- align=right
|- align=right
|7||469||730||1156||1714||2442||3379||4563||6034||7835||10012
|7||469||730||1100||1606||2278||3149||4255||5635||7331||9388
|- align=right
|8||758||1128||1634||2306||3177||4283||5663||7359||9416||11882
|- align=right
|9||1164||1670||2342||3213||4319||5699||7395||9452||11918||14844
|}
|}




<table width=100%><tr>

<td>
{| border="1"
{| border="1" align=center cellspacing=0
|+ Dimensions of chord diagram spaces with 3 chords
|+ Dimensions of chord diagram spaces with 3 chords
!rowspan="2" width="100px"|Number of Lines
!rowspan="2" width="100px"|Number of Lines
!colspan="7"|Number of Circles
!colspan="7"|Number of Circles
|- align=right
⚫
|-

|0||1||2||3||4||5||6
|0||1||2||3||4||5||6
|- align=right
|- align=right
|0||0||3||19||92||370||1120||2778
|0||0||3||19||80||270||770||1918
|- align=right
|- align=right
|1||3||19||88||351||1096||2768||6083
|1||3||19||80||270||770||1918||4284
|- align=right
|- align=right
|2||23|| 88||329||1053||2734||6073||12176
|2||23||88||283||789||1944||4318||8803
|- align=right
|3||111||321||845||2021||4419||8931||16876
|- align=right
|4||394||954||2172||4618||9184||17189
|- align=right
|5||1130||2418||4944||9600||17705||31125
⚫
|}
</td><td>
{| border="1" align=center cellspacing=0
|+ Dimensions of chord diagram spaces with 4 chords
!rowspan="2" width="100px"|Number of Lines
!colspan="6"|Number of Circles
|- align=right
|- align=right
|3||111||321||991||2657||6029||12166||22689
|0||1||2||3||4
|- align=right
|- align=right
|4||394||954||2524||5908||12112||22679
|0||0|| 6||44||241||1063
|- align=right
|- align=right
|5||1130||2418||5664||11937||22615||39875
|1||6||44||241||1063||3930
|- align=right
|- align=right
|6||2778||5424||11533||22376||39801||66793
|2||60||283||1160||4126
|}
|}
</td></tr></table>

Latest revision as of 06:35, 10 October 2006

This page documents some computational results obtained by using the ChordsMod4T software component of the VasCalc project. We denote by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal A}_n(l,m)} the formal (rational) vector space generated by all chord diagrams with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} chords on a skeleton of lines and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} circles, modulo the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4T} relations. The algorithm is straightforward: first all such diagrams are constructed, then relations are introduced iteratively over all possible terms in a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4T} relation. See the documentation for further information about the program and how these results were generated.

Some Dimensions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal A}_n(l,m)}

The tables below give the dimensions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal A}_n(l,m)} for various values of , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} . Each table corresponds to a fixed value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} - that is, the number of chords.

This is work in progress - these numbers are not reliable yet!!!

Dimensions of chord diagram spaces with 2 chords
Number of Lines Number of Circles
0 1 2 3 4 5 6 7 8 9
0 0 2 8 24 59 125 237 413 674 1044
1 2 8 24 59 125 237 413 674 1044 1550
2 9 25 60 126 238 414 675 1045 1551 2223
3 28 63 129 241 417 678 1048 1554 2226 3097
4 69 135 247 423 684 1054 1560 2232 3103 4209
5 145 257 433 694 1064 1570 2242 3113 4219 5599
6 272 448 709 1079 1585 2257 3128 4234 5614 7310
7 469 730 1100 1606 2278 3149 4255 5635 7331 9388
8 758 1128 1634 2306 3177 4283 5663 7359 9416 11882
9 1164 1670 2342 3213 4319 5699 7395 9452 11918 14844


Dimensions of chord diagram spaces with 3 chords
Number of Lines Number of Circles
0 1 2 3 4 5 6
0 0 3 19 80 270 770 1918
1 3 19 80 270 770 1918 4284
2 23 88 283 789 1944 4318 8803
3 111 321 845 2021 4419 8931 16876
4 394 954 2172 4618 9184 17189
5 1130 2418 4944 9600 17705 31125
Dimensions of chord diagram spaces with 4 chords
Number of Lines Number of Circles
0 1 2 3 4
0 0 6 44 241 1063
1 6 44 241 1063 3930
2 60 283 1160 4126
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