A note on the nearly additivity of knot width

Abstract

Let k be a knot in S 3 . In [8], H.N. Howards and J. Schultens introduced a method to construct a manifold decomposition of double branched cover of (S 3 , 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 (S 3 , 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.

Rectangle Condition for Compression Bodies and 2-Fold Branched Coverings

“…In [5], the first author gave some condition for knots satisfying the additivity of width via 2-fold branched covering as discussed in Section 5. Theorem 6.1 ([5]).…”