Thurston norm minimizing surfaces and skein trees for links in \(S^ 3\). (English) Zbl 0686.57004

Thompson, Abigail

Proc. Am. Math. Soc. 106, No. 4, 1085-1094 (1989).

The paper shows how a minimal genus spanning surface (or more properly, a spanning surface with minimal Thurston norm) can be built up for any link through a sequence of surfaces starting from a disc and changing the norm by at most one at each step. The steps used are either twisting about an arc in the spanning surface, or plumbing a Hopf band to that surface.
The process is closely connected with an analysis of the skein tree for the link, and uses results of M. Scharlemann and A. Thompson [Link genus and the Conway moves, Comment. Math. Helv. (to appear)] about the Thurston norm of adjacent nodes in the tree. A consequence of that paper is a bound on the height of the skein tree of a link L in terms of the minimal Thurston norm for a surface spanning L. The results in this paper show that when the bound is realized, the link then arises by successively plumbing Hopf bands to a disc, and is hence a fibred link.

Reviewer: H.Morton

MSC:

57M25

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

Keywords:

skein tree for a link; minimal genus spanning surface; spanning surface with minimal Thurston norm; plumbing Hopf bands; fibred link