VasCalc Results - ChordMod4T

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 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}(l,m,n)} for various values 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} , 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} 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 n} . Each table corresponds to a fixed value 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 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