Extended complexity of 3-manifolds
Sergei Matveev
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Any compact 3-manifold $M$ has a complexity
$c(M)$, which is a nonnegative integer number and which is defined as the
number of true vertices of a minimal almost simple spine $P$ of $M$.
The complexity has many good properties. In particular, it behave well
with respect to cutting $M$ along surfaces. Namely, if $M_F$ is
obtained from $M$ by cutting along an incompressible surface $F\subset M$,
then $c(M_F)\leq c(M)$. However, this useful property has a shortcoming:
the inequality is not strong. So we cannot use it for inductive proofs. We improve that
by defining extended complexity $\bar c(M)$. It is not a number anymore, but
a finite tuple of nonnegative integers.
The tuples are considered in lexicographical ordering. We prove that if $F$ is essential,
then $\bar c(M_F)< \bar c(M)$. We apply the extended complexity
for proving the algorithmic classification theorem for Haken 3-manifolds.
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