Finding disjoint Seifert surfaces. (English) Zbl 0654.57005

Scharlemann, Martin; Thompson, Abigail

Bull. Lond. Math. Soc. 20, No. 1, 61-64 (1988).

The authors prove that given two orientable surfaces S, T of minimal genus, spanning a tubular neighborhood of a knot K, there is a sequence of surfaces \(S=S_ 0,S_ 1,S_ 2...S_{n-1},S_ n=T\), such that \(S_ i\) and \(S_{i-1}\) are disjoint, and the \(S_ i\) are spanning surfaces of minimal genus for K. It was not previously known that the intervening surfaces could be all of minimal genus.
They also show there is a sequence of spanning surfaces starting with an arbitrary orientable surface, ending with a surface of minimal genus, and with each successive surface of lesser genus than and disjoint from the preceding.

Reviewer: L.Neuwirth

MSC:

57M25

Knots and links in the \(3\)-sphere (MSC2010)

Keywords:

two orientable surfaces of minimal genus; Seifert surfaces of knots; knot; spanning surfaces