View source for VasCalc Results - ChordMod4T

This page documents some computational results obtained by using the <b>ChordsMod4T</b> software component of the [[VasCalc]] project. We denote by <math>{\mathcal A}(l,m,n)</math> the formal (rational) vector space generated by all chord diagrams with <math>n</math> chords on a skeleton of <math>l</math> lines and <math>m</math> circles, modulo the <math>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>4T</math> relation. See the [[VasCalc Documentation - ChordsMod4T| documentation]] for further information about the program and how these results were generated.

===Some Dimensions of <math>{\mathcal A}(l,m,n)</math>===

The tables below give the dimensions of <math>{\mathcal A}(l,m,n)</math> for various values of <math>l</math>, <math>m</math> and <math>n</math>. Each table corresponds to a fixed value of <math>n</math> - that is, the number of chords. 

'''This is work in progress - these numbers are not reliable yet!!!'''

{| border="1"
|+ Dimensions of chord diagram spaces with 2 chords
!rowspan="2" width="100px"|Number of Lines
!colspan="10"|Number of Circles
|- align=right
||0||1||2||3||4||5||6||7||8||9
|- align=right
|0||0|| 2||8||24||59||125||237||413||674||1044
|- align=right
|1||2||8||24||59||125||237||413||674||1044||1550
|- align=right
|2||9||25||60||126||238||414||675||1045||1551||2223
|- align=right
|3||28||63||129||241||417||678||1048||1554||2226||3097
|- align=right
|4||69||135||247||423||684||1054||1560||2232||3103||4209
|- align=right
|5||145||257||433||694||1064||1570||2242||3113||4219||5599
|- align=right
|6||272||448||709||1079||1585||2257||3128||4234||5614||7310
|- align=right
|7||469||730||1100||1606||2278||3149||4255||5635||7331||9388
|- align=right
|8||758||1128||1634||2306||3177||4283||5663||7359||9416||11882
|- align=right
|9||1164||1670||2342||3213||4319||5699||7395||9452||11918||14844
|}



{| border="1" 
|+ Dimensions of chord diagram spaces with 3 chords
!rowspan="2" width="100px"|Number of Lines
!colspan="7"|Number of Circles
|- align=right

|0||1||2||3||4||5||6
|- align=right
|0||0||3||19||80||270||770||1918
|- align=right
|1||3||19||80||270||770||1918||4284
|- align=right
|2||23||88||283||789||1944||4318||8803
|- align=right
|3||111||321||845||2021||4419||8931||16876
|- align=right
|4||394||954||2172||4618||9184||17189
|- align=right
|5||1130||2418||4944||9600||17705||31125
|}