Circular handle decompositions of free genus one knots

Fabiola Manjarrez-Gutiérrez, Víctor Núñez and Enrique Ramírez-Losada

Vol. 275 (2015), No. 2, 361–407

DOI: 10.2140/pjm.2015.275.361

Abstract

We determine the structure of the circular handle decompositions of the family of free genus one knots. Namely, if $k$ is a free genus one knot, then the handle number $h\left(k\right)=$ 0, 1 or 2, and, if $k$ is not fibered (that is, if $h\left(k\right)>0$), then $k$ is almost fibered. For this, we develop practical techniques to construct circular handle decompositions of knots with free Seifert surfaces in the 3-sphere (and compute handle numbers of many knots), and, also, we characterize the free genus one knots with more than one Seifert surface. These results are obtained through analysis of spines of surfaces on handlebodies. Also we show that there are infinite families of free genus one knots with either $h\left(k\right)=1$ or $h\left(k\right)=2$.

Keywords

circular thin position, free genus, free genus one knots, Seifert surfaces, handle decompositions, almost fibered

Mathematical Subject Classification 2010

Primary: 57M25

Milestones

Received: 14 January 2014

Revised: 28 April 2014

Accepted: 28 April 2014

Published: 15 May 2015