8 SELMA ALTINOK AND MOHAN BHUPAL
that ki = ki ki = ki for i = 1, . . . , l. Thus, by Lemma 3.1, the page-genus of
OB(Z(T )) is given by
g(Z(T )) = 1 +
l
i=1
(wi 2)zi + (zi 1)ki
2
.
Substituting vi = wi + ti into (3.3), where ti is the number of non-Tjurina
components intersecting Ei, we obtain
g(Y + Z(T )) g(Y ) =
l
i=1
(wi + ti + ki 2)zi ki
2

n
i=l+1
ki ki
2
= g(Z(T )) 1 +
l
i=1
tizi
2

n
i=l+1
ki ki
2
= g(Z(T )) 1 +
l
i=1
tizi
2
+
n
i=l+1
zφ(i)
2
= g(Z(T )) 1 +
l
i=1
tizi,
where the penultimate equality follows from the fact that ki ki = zφ(i) for i =
l + 1, . . . , n, which in turn follows from (3.2). Now, appealing to Theorem 3.10 in
[12], we see that zi = 1 whenever ti = 0. This completes the proof.
Lemma 3.3. Let Y =
∑n
i=1
miEi be an element of E + and Ei be a non-Tjurina
component for Y that does not intersect any Tjurina component for Y . Then
genus(OB(Y + Ei)) = genus(OB(Y )) + vi Y · Ei 1.
Proof. For convenience we will assume that E1 is our non-Tjurina component.
Define ki, for i = 1, . . . , n, by (3.1) and define ki by replacing (m1, . . . , mn) by
(m1 + 1, m2, . . . , mn) in (3.1). Arguing as in the proof of Lemma 3.2, we now
obtain that
g(Y + E1) g(Y ) =
(v1 + k1 + k1 2) (k1 k1)
2

n
i=2
ki ki
2
=
v1 + 2k1 2
2
+
v1
2
= v1 + k1 1,
where the second equality follows from the fact that ki ki = 1 for the indices
of the v1 curves intersecting E1, and ki ki = 0 for all the other indices. This
completes the proof.
Proof of Theorem 1.1. Use Theorem 2.6, Lemma 3.2, Lemma 3.3 and Re-
mark 2.8.
Proof of Theorem 1.2. First note that the number of binding components
of an open book OB(Y ) for Y E
+
is given by
bc(OB(Y )) =
n
i=1
Y · Ei.
8
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