Narrow Results

The splice quotients are an interesting class of normal surface singularities with rational homology sphere links. In general, it can be difficult to determine whether or not a singularity is a splice quotient (an analytic condition). We consider splice quotient deformations of splice quotients of the form z² = x^a + y^b, and show that in general not all equisingular deformations are splice quotients.

A lot of attention has been paid recently to the construction of symmetric covers of symmetric graphs. After a new approach given by Conder and the author [Arc-transitive abelian regular covers of cubic graphs, J. Algebra387 (2013) 215–242], the group of covering transformations can be extended to more general abelian groups rather than cyclic or elementary abelian groups. In this paper, by using the Conder–Ma approach, we investigate the symmetric covers of 4-valent symmetric graphs. As an application, all the arc-transitive abelian regular covers of the 4-valent complete graph $K_5$ which can be obtained by lifting the arc-transitive subgroups of automorphisms $A_5$ and $\mathrm{\text{A}}GL(1,5)$ are classified.

Let M be the manifold obtained by 0-framed surgery along a knot K in the 3-sphere. A Topological Quantum Field Theory assigns to a fundamental domain of the universal abelian cover of M an operator, whose non-nilpotent part is the Turaev-Viro module of K. In this paper, using surgery formulas, we give a matrix presentation for the Turaev-Viro module of any knot K, in the case of the (V_p, Z_p) TQFT of Blanchet, Habegger, Masbaum and Vogel. We do the computation for a family of knots in the special case p = 8, and note the relation with the fibering question.

We describe how a coarse classification of graph manifolds can give clearer insight into their structure, and we relate this particularly to the manifolds that can occur as the links of points in normal complex surfaces. We relate this discussion to a special class of singularities; those of “splice type”, which turn out to play a central role among singularities of complex surfaces.

An appendix gives a brief introduction to the relevant parts of classical 3-manifold theory.