VasCalc Results - ChordMod4T
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!!!
| 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 |
| 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 | |
