VasCalc Results - ChordMod4T: Difference between revisions

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Revision as of 16:01, 7 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.

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 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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