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!!!'''
'''This is work in progress - these numbers are not reliable yet!!!'''
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<table width=100%><tr>

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

|0||1||2||3||4||5||6
|0||1||2||3||4||5||6
|- align=right
|- align=right
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|5||1130||2418||4944||9600||17705||31125
|5||1130||2418||4944||9600||17705||31125
|}
|}
</td><td>


{| border="1" align=center cellspacing=0
{| border="1" align=center cellspacing=0
|+ Dimensions of chord diagram spaces with 4 chords
|+ Dimensions of chord diagram spaces with 4 chords
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!colspan="6"|Number of Circles
!colspan="6"|Number of Circles
|- align=right
|- align=right

|0||1||2||3||4
|0||1||2||3||4
|- align=right
|- align=right
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|- align=right
|- align=right
|2||60||283||1160||4126
|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 [math]\displaystyle{ {\mathcal A}_n(l,m) }[/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}_n(l,m) }[/math]

The tables below give the dimensions of [math]\displaystyle{ {\mathcal A}_n(l,m) }[/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.

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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