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	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	5
0	0	6	44	241	1063
1	6	44	241	1063	3930
2	60	283	1160	4126