A note on the nearly additivity of knot width

TL;DR

This paper proves that under certain conditions, the knot width is nearly additive when a knot is decomposed into prime summands, and generalizes the result to cases involving topologically minimal surfaces.

Contribution

It establishes a nearly additivity property of knot width for knots with thin positions inducing specific manifold decompositions, extending previous results.

Findings

Knot width is nearly additive for certain thin positions.

Thin positions can be decomposed into prime summands with additive properties.

Generalization to topologically minimal surfaces broadens applicability.

Abstract

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.)