On a conjecture of M. J. Dunwoody

We deal with three combinatorial representations of closed orientable 3-manifolds, i.e., Heegaard diagrams, branched coverings, and crystallizations (a special class of pseudo-graphs endowed with proper edge-colorings). Exploring the connections between those theories, we prove the validity of a conjecture, stated by Dunwoody in Groups-Korea 1994, Walter de Gruyter, 1995, concerning the class of closed orientable 3-manifolds represented by symmetric Heegaard diagrams. As a consequence, we classify the topological and geometric structures of many interesting classes of cyclic branched coverings of (hyperbolic) knots encoded by cyclic presentations of groups. In all cases, we show that the polynomial associated with the cyclic presentation coincides (up to a multiplicative unit) with the Alexander polynomial of the considered knot. Finally, we include a partial output of a computer program which generates symmetric Heegaard diagrams of cyclic branched coverings of 3-bridge knots up to nine crossings.

We deal with three combinatorial representations of closed orientable 3-manifolds, i.e., Heegaard diagrams, branched coverings, and crystallizations (a special class of pseudo-graphs endowed with proper edge-colorings). Exploring the connections between those theories, we prove the validity of a conjecture, stated by Dunwoody in [14], concerning the class of closed orientable 3-manifolds represented by symmetric Heegaard diagrams. As a consequence, we classify the topological and geometric structures of many interesting classes of cyclic branched coverings of (hyperbolic) knots encoded by cyclic presentations of groups. In all cases, we show that the polynomial associated with the cyclic presentation coincides (up to a multiplicative unit) with the Alexander polynomial of the considered knot. Finally, we include a partial output of a computer program which generates symmetric Heegaard diagrams of cyclic branched coverings of 3-bridge knots up to nine crossings. © Inst. Math. CAS 2001.