Difference between revisions of "VasCalc Results - ChordMod4T"

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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>
{| border="1"  
+
<td>
 +
{| 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 07:35, 10 October 2006

This page documents some computational results obtained by using the ChordsMod4T software component of the VasCalc project. We denote by {\mathcal A}_n(l,m) 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 4T relation. See the documentation for further information about the program and how these results were generated.

Some Dimensions of {\mathcal A}_n(l,m)

The tables below give the dimensions of {\mathcal A}_n(l,m) for various values of l, m and n. Each table corresponds to a fixed value of 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