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 A(l,m,n) the formal (rational) vector space generated by all chord diagrams with n chords on a skeleton of l lines and m circles, modulo the 4T relations. The algorithm is straightforward: first all such diagrams are constructed, then relations are introduced iteratively over all possible terms in a 4-T 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}(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.


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


The tables below give the dimensions of A(l,m,n) for various values of l, m and n. Each table corresponds to a fixed value of n - that is, the number of chords.
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.




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!rowspan="2" width="100px"|Number of Lines
!rowspan="2" width="100px"|Number of Lines
!colspan="10"|Number of Circles
!colspan="10"|Number of Circles
|-
|- align=right
||0||1||2||3||4||5||6||7||8||9
||0||1||2||3||4||5||6||7||8||9
|- align=right
|-
|0||0||2||8||27||69||145||272||469||758||1164
|0||0||2||8||27||69||145||272||469||758||1164
|- align=right
|-
|1||2||8||26||68||145||272||469||758||1164||1715
|1||2||8||26||68||145||272||469||758||1164||1715
|- align=right
|-
|2||9||25||66||144||272||469||758||1164||1715||2442
|2||9||25||66||144||272||469||758||1164||1715||2442
|- align=right
|-
|3||28||63||141||271||469||758||1164||1715||2442||3379
|3||28||63||141||271||469||758||1164||1715||2442||3379
|- align=right
|-
|4||69||135||267||468||758||1164||1715||2442||3379||4563
|4||69||135||267||468||758||1164||1715||2442||3379||4563
|- align=right
|-
|5||145||257||463||757||1164||1715||2442||3379||4563||6034
|5||145||257||463||757||1164||1715||2442||3379||4563||6034
|- align=right
|-
|6||272||448||751||1163||1715||2442||3379||4563||6034||7835
|6||272||448||751||1163||1715||2442||3379||4563||6034||7835
|- align=right
|-
|7||469||730||1156||1714||2442||3379||4563||6034||7835||10012
|7||469||730||1156||1714||2442||3379||4563||6034||7835||10012
|}
|}
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|0||1||2||3||4||5||6
|0||1||2||3||4||5||6
|- align=right
|-
|0||0||3||19||92||370||1120||2778
|0||0||3||19||92||370||1120||2778
|- align=right
|-
|1||3||19||88||351||1096||2768||6083
|1||3||19||88||351||1096||2768||6083
|- align=right
|-
|2||23|| 88||329||1053||2734||6073||12176
|2||23|| 88||329||1053||2734||6073||12176
|- align=right
|-

|3||111||321||991||2657||6029||12166||22689
|3||111||321||991||2657||6029||12166||22689
|- align=right
|-
|4||394||954||2524||5908||12112||22679
|4||394||954||2524||5908||12112||22679
|- align=right
|-
|5||1130||2418||5664||11937||22615||39875
|5||1130||2418||5664||11937||22615||39875
|- align=right
|-
|6||2778||5424||11533||22376||39801||66793
|6||2778||5424||11533||22376||39801||66793
|}
|}

Revision as of 08:40, 4 July 2006

This page documents some computational results obtained by using the ChordsMod4T software component of the VasCalc project. We denote by [math]\displaystyle{ {\mathcal A}(l,m,n) }[/math] the formal (rational) vector space generated by all chord diagrams with [math]\displaystyle{ n }[/math] chords on a skeleton of [math]\displaystyle{ l }[/math] lines and [math]\displaystyle{ m }[/math] circles, modulo the [math]\displaystyle{ 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]\displaystyle{ 4T }[/math] relation. See the documentation for further information about the program and how these results were generated.

Some Dimensions of [math]\displaystyle{ {\mathcal A}(l,m,n) }[/math]

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


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 27 69 145 272 469 758 1164
1 2 8 26 68 145 272 469 758 1164 1715
2 9 25 66 144 272 469 758 1164 1715 2442
3 28 63 141 271 469 758 1164 1715 2442 3379
4 69 135 267 468 758 1164 1715 2442 3379 4563
5 145 257 463 757 1164 1715 2442 3379 4563 6034
6 272 448 751 1163 1715 2442 3379 4563 6034 7835
7 469 730 1156 1714 2442 3379 4563 6034 7835 10012


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 92 370 1120 2778
1 3 19 88 351 1096 2768 6083
2 23 88 329 1053 2734 6073 12176
3 111 321 991 2657 6029 12166 22689
4 394 954 2524 5908 12112 22679
5 1130 2418 5664 11937 22615 39875
6 2778 5424 11533 22376 39801 66793
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