Thin position for 3-manifolds was defined by M. Scharlemann and A. Thompson [Contemp. Math. 164, 231–238 (1994; Zbl 0818.57013)] by adapting the concept of thin position for knots in \(S^3\) developed by D. Gabai. F. Manjarrez-Gutiérrez [Algebr. Geom. Topol. 9, No. 1, 429–454 (2009; Zbl 1171.57005)] developed curcular width and circular thin position of the knot exterior based on the idea of thin position for 3-manifolds. A fibered knot means that its knot exterior has a circular thin position with one and only one incompressible level surface. An almost fibered knot means that its knot exterior has a circular thin position where there is only one weakly incompressible Seifert surface and one incompressible Seifert surface. F. Manjarrez-Gutiérrez et al. in [Pac. J. Math. 275, No. 2, 361–407 (2015; Zbl 1321.57011)] proved that if a non-fibered knot is free genus one then it is an almost-fibered knot. This implies that any knot has a circular decomposition with only one thick surface. Examples of non almost-fibered knots make the other direction of this topic.
In this paper, the authors explicitly construct an infinite family of genus one hyperbolic knots which are not almost-fibered. For this construction, they ideally consider four non-isotopic genus one Seifert surfaces; two of these are thin surfaces of genus one and the other two are thick surfaces of genus two of a decomposition of the knot exterior. They prove that the example they construct cannot have a decomposition with only one 1-handle and one 2-handle. Therefore, they are not almost-fibered.