  Let k be a knot in S3. In [8], H.N. Howards and J. Schultens introduced a
method to construct a manifold decomposition of double branched cover of (S3,
k) from a thin position of k. In this article, we will prove that if a thin
position of k induces a thin decomposition of double branched cover of (S3,k)
by Howards and Schultens' method, then the thin position is the sum of prime
summands by stacking a thin position of one of prime summands of k on top of a
thin position of another prime summand, and so on. Therefore, k holds the
nearly additivity of knot width (i.e. for k = k1#k2, w(k) = w(k1)#w(k2) - 2) in
this case. Moreover, we will generalize the hypothesis to the property a thin
position induces a manifold decomposition whose thick surfaces consists of
strongly irreducible or critical surfaces (so topologically minimal.)
