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 the formal (rational) vector space generated by all chord diagrams with chords on a skeleton of lines and circles, modulo the relations. The algorithm is straightforward: first all such diagrams are constructed, then relations are introduced iteratively over all possible terms in a relation. See the documentation for further information about the program and how these results were generated.

Some Dimensions of

The tables below give the dimensions of for various values of , and . Each table corresponds to a fixed value of - 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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