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}_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!!!

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 80 270 770 1918 1 3 19 80 270 770 1918 4284 2 23 88 283 789 1944 4318 8803 3 111 321 845 2021 4419 8931 16876 4 394 954 2172 4618 9184 17189 5 1130 2418 4944 9600 17705 31125

Dimensions of chord diagram spaces with 4 chords Number of Lines Number of Circles 0 1 2 3 4 0 0 6 44 241 1063 1 6 44 241 1063 3930 2 60 283 1160 4126