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!!!

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