Genus one 1-bridge knots and Dunwoody manifolds (Q2716423)

In [Groups -- Korea '94, 47-55 (1995; Zbl 0871.20026)]\textit{M. J. Dunwoody} considers a class of planar graphs with cyclic symmetries which define Heegaard diagrams of a class of closed orientable 3-manifolds (``Dunwoody manifolds''). The presentations of the fundamental groups associated to the Heegaard diagrams of these 3-manifolds are cyclic, i.e. the relations are obtained from a single word in \(n\) generators by cyclically permuting the generators in the word. In the present paper, these manifolds and their induced cyclic symmetries are studied in detail. The main result states that these manifolds are \(n\)-fold cyclic coverings branched over genus one 1-bridge knots in lens spaces (including \(S^3\) and \(S^2 \times S^1\)), giving also a positive answer to a question of Dunwoody in this regard (recall that a link in a 3-manifold has a genus \(g\) \(b\)-bridge presentation if the link intersects each handlebody of a genus \(g\) Heegaard decomposition of the 3-manifold in a system of \(b\) trivial arcs). It is shown that all 2-bridge knots in the 3-sphere occur among the branch sets, so all cyclic branched coverings of 2-bridge knots are Dunwoody manifolds and their fundamental groups admit cyclic geometric presentations (i.e. associated to Heegaard diagrams). The authors conjecture that also all torus knots occur among the branch sets, and consequently that also their cyclic branched coverings admit cyclic geometric presentations.