Abstract
A rational Ansatz is proposed for the generating function
∑
j,k
β
2j+k,2j
x
j
y
k
, where
β
m,u
is the number of primitive chinese character diagrams with
u
univalent and
2m−u
trivalent vertices. For
P
m
:=
∑
u≥2
β
m,u
, the conjecture leads to the sequence
1,1,1,2,3,5,8,12,18,27,39,55,
78,108,150,207,284,388,532,726
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
for primitive chord diagrams of degrees
m≤20
, with predictions underlined. The asymptotic behaviour
lim
m→∞
P
m
/
r
m
=1.06260548918755
results, with
r=1.38027756909761
solving
r
4
=
r
3
+1
. Vassiliev invariants of knots are then enumerated by
0,1,1,3,4,9,14,27,44,80,132,232,
384,659,1095,1851,3065,5128,8461,14031
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
and Vassiliev invariants of framed knots by
1,2,3,6,10,19,33,60,104,184,316,548,
932,1591,2686,4537,7602,12730,21191,35222
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
These conjectures are motivated by successful enumerations of irreducible Euler sums. Predictions for
β
15,10
,
β
16,12
and
β
19,16
suggest that the action of sl and osp Lie algebras, on baguette diagrams with ladder insertions, fails to detect an invariant in each case.