A NULLSTELLENSATZ FOR AMOEBAS

SOME ASYMPTOTICS OF TQFT VIA SKEIN THEORY 581
which the mapping class group acts are a priori completely different. We need a way
to compare them; this is suggested by Proposition 2.3.
Given a multicurve γ in , one can give it a banded structure by taking a
neighborhood of it in . We can consider the curve γ as a banded link in × [0, 1]
by sending it to γ ×{1/2}. We use the same notation for the multicurve on and its
associated banded link in × [0, 1].
In [PS], it is shown that the Kauffman skein module K( × [0, 1]) is a free
Z[A, A −1 ]-module with a basis of the isotopy classes of multicurves. It provides an
isomorphism of vector spaces between C() and K( ×[0, 1],u) for any u in C\{0}.
In particular, using Proposition 2.3, we get a surjective map
ϕ p : C() ∼ → K( × [0, 1], −e iπ/p ) → V p (∐ − ).
THEOREM 3.1
Let be a closed oriented surface of genus g. There is a Hermitian pairing 〈·, ·〉 on
C() such that for all x and y in C(), the following holds, where d(1) = 1 and
d(g) = 3g − 3 for g>1:
2
) d(g)〈ϕp
〈x,y〉= lim (x),ϕ
p→∞(
p (y)〉 p .
p
3.2. The trace function
Definition 3.2
Let be a closed oriented surface of genus g, andletγ be a multicurve on . We
set Tr p (γ ) =〈 × S 1 ,γ〉 p . Here, γ is seen as a banded link with color 1 lying in the
slice ×{1/2} of × S 1 .
LEMMA 3.3
Suppose that a surface is presented as the boundary of a handlebody H which
retracts on a trivalent banded graph G as in Theorem 2.1. We choose meridian discs
D e transverse to each edge of G and define C e = ∂D e ; the curves C e are disjoint on
. We choose a nonnegative integer m e for each edge of G.
Then we define γ as the multicurve on obtained by taking m e parallel copies
of C e for each edge of G. We have
Tr p (γ ) = ∑ σ
∏
e
[ ( (σe + 1)π
−2 cos
r
)] me.
Here, σ ranges over r-admissible colorings of G, and e ranges over edges of G.