Cacti, braids and complex polynomials.
Coloured knots and permutations representing 3-manifolds.
Combinatorial mapping-torus, branched surfaces and free group automorphisms
We give a characterization of the geometric automorphisms in a certain class of (not necessarily irreducible) free group automorphisms. When the automorphism is geometric, then it is induced by a pseudo-Anosov homeomorphism without interior singularities. An outer free group automorphism is given by a -cocycle of a -complex (a standard dynamical branched surface, see [7] and [9]) the fundamental group of which is the mapping-torus group of the automorphism. A combinatorial construction elucidates...
Complete Intersections as Branched Covers and the Kervaire Invariant.
Corrigendum et addendum ad: “Minimal cell coverings of sphere bundles over spheres”
Covering homotopy 3-spheres.
Coverings defined by Weierstraß polynomials.
Coverings of S3 branched over iterated torus links.
Coverings of S3 branched over iterated torus links appear naturally and very often in Algebraic Geometry. The natural graph-manifold structure of the exterior of an iterated torus link induces a graph-structure in the branched covers. In this paper we give an algorithm to compute valued graphs representing a branched cover given the monodromy representation associated to the covering. The algorithm is completely mechanized in order to be programmed, and can also be used for finding representation...
Cyclic actions on - and -bundles over
Cyclic branched coverings and homology 3-spheres with large group actions
We show that, if the covering involution of a 3-manifold M occurring as the 2-fold branched covering of a knot in the 3-sphere is contained in a finite nonabelian simple group G of diffeomorphisms of M, then M is a homology 3-sphere and G isomorphic to the alternating or dodecahedral group 𝔸₅ ≅ PSL(2,5). An example of such a 3-manifold is the spherical Poincaré sphere. We construct hyperbolic analogues of the Poincaré sphere. We also give examples of hyperbolic ℤ₂-homology 3-spheres with PSL(2,q)-actions,...
Cyclic branched coverings of 2-bridge knots.
In this paper we study the connections between cyclic presentations of groups and the fundamental group of cyclic branched coverings of 2-bridge knots. Then we show that the topology of these manifolds (and knots) arises, in a natural way, from the algebraic properties of such presentations.
Cyclic branched coverings of knots and homology spheres.
We study cyclic coverings of S3 branched over a knot, and study conditions under which the covering is a homology sphere. We show that the sequence of orders of the first homology groups for a given knot is either periodic of tends to infinity with the order of the covering, a result recently obtained independently by Riley. From our computations it follows that, if surgery on a knot k with less than 10 crossings produces a manifold with cyclic fundamental group, then k is a torus knot.