MINIMAL PAGE-GENUS OF MILNOR OPEN BOOKS 5
of circle bundles over spheres with disjoint collections of discs removed. The genera
of the intersections of these with a page of OB(Y ) is the first contribution we
consider. The second contribution arises from the way these various pieces fit
together and is more global in nature. We will consider these two contributions
separately. We begin with the first contribution.
Assume, without loss of generality, that the irreducible curves Ei of E are
ordered so that Ei intersects some curve in {E1, . . . , Ei−1} for each i = 2, . . . , n.
Define nonnegative integers ki, for i = 1, . . . , n, by
(3.1) (m1, . . . , mn)IE = (−k1, . . . , −kn),
where IE is the intersection matrix of E. Thus ki = −Y · Ei for each i. Let
di =
gcd({mi} {mj | Ej · Ei = 1}) if ki = 0,
1 otherwise
for i = 1, . . . , n. Now consider the open book OB(Y ) associated to the element
Y =
∑n
i=1
miEi E +. The part of a page Σ of OB(Y ) associated to Ei, that is,
the intersection of Σ with the corresponding circle bundle over a punctured sphere
associated with Ei, splits into di connected components Fis, s = 1, . . . , di, each with
genus gi given by
gi = 1 +
(vi + ki 2)mi/di

j s.t. Ei · Ej = 1
(mi, mj )/di ki
2
.
(See [2] for an explanation of the formula for gi.) Thus the first contribution to the
page-genus of OB(Y ) is given by
I =
n
i=1
digi =
n
i=1
di +
(vi + ki 2)mi

(mi, mj ) ki
2
,
where the summation in the numerator of the last expression is taken over all j
such that Ei · Ej = 1. In the last line we use the equality kidi = ki. This follows
from the fact that if ki = 0, then di = 1.
Before considering the second contribution we observe the following: Suppose
that Eµ, are two irreducible curves in E such that · = 1. Then notice
that the corresponding parts =

s=1

s
and =

t=1

t
of a page Σ of the
open book OB(Y ) split into collections Fµ,k and Fν,, for k, = 1, . . . , (dµ, ), of
dµ/(dµ, ) and /(dµ, ) components respectively such that on plumbing only
boundary components of Fµ,k and Fν,k, for k = 1, . . . (dµ, ), are connected by
annuli. Also, since there are (mµ, ) annuli connecting to in Σ, there must
be (mµ, )(dµ, )/dµdν annuli connecting each connected component of Fµ,k to
each connected component of Fν,k for k = 1, . . . , (dµ, ).
We now return to the consideration of the second contribution. We proceed by
building up a page Σ of OB(Y ) by starting with the part of Σ corresponding to E1
and then, at each stage, for i = 2, . . . , n, attaching the part of Σ corresponding to
Ei to the part of Σ corresponding to E1 ··· Ei−1 by a set of annuli that arises
from the corresponding plumbing. For each i {2, . . . , n}, let φ(i) {1, . . . , i 1}
be the unique index such that Eφ(i) · Ei = 1. There are three parts to the second
contribution. The first part arises as a consequence of the possibility that there may
be more than one annulus connecting connected components of Fi,k to connected
5
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