VasCalc Results - ChordMod4T

From Drorbn
Revision as of 09:40, 4 July 2006 by Drorbn (Talk | contribs)

Jump to: navigation, search

This page documents some computational results obtained by using the ChordsMod4T software component of the VasCalc project. We denote by {\mathcal A}(l,m,n) 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}(l,m,n)

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


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 27 69 145 272 469 758 1164
1 2 8 26 68 145 272 469 758 1164 1715
2 9 25 66 144 272 469 758 1164 1715 2442
3 28 63 141 271 469 758 1164 1715 2442 3379
4 69 135 267 468 758 1164 1715 2442 3379 4563
5 145 257 463 757 1164 1715 2442 3379 4563 6034
6 272 448 751 1163 1715 2442 3379 4563 6034 7835
7 469 730 1156 1714 2442 3379 4563 6034 7835 10012


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