Free Seifert surfaces and disk decompositions

Abstract

A Seifert surface F for a knot K is disk decomposable if there is a taut sutured manifold heirarchy for the complement of F, whose decomposing surfaces are all disks. It follows that F has minimal genus for the knot K, and has handlebody complement, i.e., F is free. We show that these necessary conditions for disk decomposability are not sufficient, by constructing a family of knots with genus one free Seifert surfaces, which are not disk decomposable.

14 pages, with 18 figures (in color). For a version with figures in black and white (which prints better), visit http://www.math.unl.edu/~mbritten/personal/pprdescr.html