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6 Operads
The component QO(S; G) is defined as
QO(S; G) :=
{c 1 , . . . , c b }
g
| b > 0, g ≥ 0,
b
i=1
c i = S, G = 2g + b − 1
,
where {c 1 , . . . , c b } G is a symbol consisting of a non-negative integer g ≥ 0 and
an (unordered) set of cycles whose disjoint union is S. The natural number g is
the geometric genus of the surface represented by a given symbol. For a bijection
ρ : S
∼ =
− → T , let
QO(ρ)({c 1 , . . . , c b }
g ) := {ρ(c 1 ), . . . , ρ(c b )}
g .
Next, we define the composition operations
a ◦ b : QO
S 1 {a}, G 1
⊗ QO
S 2 {b}, G 2
→ QO
S 1 S 2 , G 1 + G 2
.
Assume that c i = ( (a, x 1 , . . . , x m ) ) is a cycle in S 1 and let d j = ((b, y 1 , . . . , y n ))
be a cycle in S 2 {b}. Then
c 1 , . . . , c b 1 }
g 1 a ◦ b
d 1 , . . . , d b 2
g 2
:=
((x 1 , . . . , x m , y 1 , . . . , y n ) ) , c 1 , . . . ,
c i , . . . , c b 1 , d 1 , . . . ,
d j , . . . , d b 2
g 1 +g 2 .
The contractions
◦ uv = ◦ vu : QO
S {u, v}; G
→ QO(S; G + 1)
are defined as follows. Let {c 1 , . . . , c b } g ∈ QO(S, G). If there are i < j such that
c i = ( (u, x 1 , . . . , x m )) and c j = ((v, y 1 , . . . , y n ) ), then define
◦ uv ({c 1 , . . . , c b }
g ) := {((x 1 , . . . , x m , y 1 , . . . , y m ) ) , c 1 , . . . ,
c i , . . . ,
c j , . . . , c b }
g+1 .
Otherwise, there is i such that c i = ( (u, x 1 , . . . , x m , v, y 1 , . . . , y n )). Then define
◦ uv ({c 1 , . . . , c b }
g ) := {((x 1 , . . . , x m )) , ((y 1 , . . . , y n )) , c 1 , . . . ,
c i , . . . , c b }
g .
Notice that we allow repeated empty cycles to appear in {c 1 , . . . , c b } g , for
example, {(()) , ( ()) , ((3) ) , ((14) ) , ((25) )} 2 ∈ QO([5], 8). Also notice that ◦ uv can
produce empty cycles, e.g.,
◦ uv {((u) ) , ((v))}
g
= {(())}
g+1 and ◦ uv {((uv))}
g
= {(()) , (())}
g .