On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces
A Heegaard decomposition of the exterior E of a knot k is the union of a handlebody \(H_ g\) of genus g with (g-1) 2-handles which are attached along curves on \(\partial H_ g\). The Heegaard genus of k is the minimal genus of decompositions of E. In this paper, the Heegaard decompositions of genus 2 of torus knot exteriors are classified with respect to homeomorphisms. It turns out that, in general, there are three different classes which are also the isotropy classes. A similar result is obtained for Seifert fibre spaces having a disc as base and two exceptional fibres, such as the torus knot exteriors have. The proof of these classification theorems are based on two results: (1) The classification of the Nielsen equivalence classes of one relator presentations of the fundamental groups, \(<S,T | S^ pT^{-q}>\), of these manifolds which has been done by the third author and D. J. Collins. (2) The determination of the ''geometric'' one-relator presentations of the fundamental group of these manifolds. These are the presentations obtained from a genus 2 Heegaard decomposition. Moreover, the different Heegaard decompositions of genus 2 of the manifolds are geometrically described.
Heegaard decompositions of genus 2 of torus knot exteriors
Nielsen equivalence classes of one relator presentations of the fundamental groups
scientific article; zbMATH DE number 3998785
scientific article