VasCalc Results - ChordMod4T

This page documents some computational results obtained by using the ChordsMod4T software component of the VasCalc project. We denote by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal A}(l,m,n)} the formal (rational) vector space generated by all chord diagrams with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} chords on a skeleton of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l} lines and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} circles, modulo the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4T} relations. The algorithm is straightforward: first all such diagrams are constructed, then relations are introduced iteratively over all possible terms in a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4T} relation. See the documentation for further information about the program and how these results were generated.

Some Dimensions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal A}(l,m,n)}

The tables below give the dimensions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal A}_n(l,m)} for various values of $l$ (lines), $m$ (circles) and $n$ (chords). Each table corresponds to a fixed value of $n$ - 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