Circular handle decompositions of free genus one knots (Q2346532)

Thin position, see \textit{M. Scharlemann} and \textit{A. Thompson} [Contemp. Math. 164, 231--238 (1994; Zbl 0818.57013)] for a 3-manifold \(M\) with boundary, and also the width of \(M\), is defined by dividing up \(M\) in a linear fashion by an alternating sequence of incompressible and weakly incompressible surfaces, such that the sub-manifold between any two consecutive surfaces is a compression body. That is, the manifold is expressed as a thickened surface with sets of 1--handles and sets of 2--handles alternately attached to one side (maybe with sphere boundary components capped off using 3-handles). A circular handle decomposition is the same concept using a circular sequence of surfaces instead of a linear one, and gives rise to the ideas of circular thin position, circular width and circular handle number. Another form in which the definition can be understood is that a circular handle decomposition is related to a circle-valued Morse function as a handle decomposition is related to a real-valued Morse function. This paper studies circular handle decompositions particularly for free genus one knots -- that is, knots that have a genus one Seifert surface whose complement is a handlebody. Note that a knot is fibred if its complement has a circular handle decomposition with just an incompressible Seifert surface. It is almost fibred if it has a circular handle decomposition with one incompressible surface and one weakly incompressible surface. The results include a characterisation of when a free genus one knot has a unique Seifert surface in terms of spines for the surface, and calculations of the handle number for rational knots and many three-strand pretzel knots. The latter gives examples of non-fibred knots with handle number strictly smaller than their tunnel number. The paper also includes explicit methods to test whether a free Seifert surface can be used to show that a given knot is almost fibred. The final result is that a free genus one knot in \(S^3\) that is not fibred is almost fibred. The key tools used are spines of surfaces and handlebodies, and Whitehead diagrams of graphs on the boundary of a handlebody (particularly the effect on these of drilling a tunnel and of handle slides). Other important concepts used are primitive elements in and splittings of fundamental groups.