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 |
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. |
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===Some Dimensions of <math>{\mathcal A}(l,m |
===Some Dimensions of <math>{\mathcal A}_n(l,m)</math>=== |
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The tables below give the dimensions of <math>{\mathcal A}_n(l,m)</math> for various values of <math>l</math> |
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. |
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'''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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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!!!
| 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 |
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