We prove that the Witten–Reshetikhin–Turaev (WRT) SO(3) invariant of an arbitrary 3-manifold M is always an algebraic integer. Moreover, we give a rational surgery formula for the unified invariant dominating WRT SO(3) invariants of rational homology 3-spheres at roots of unity of order co-prime with the torsion. As an application, we compute the unified invariant for Seifert fibered spaces and for Dehn surgeries on twist knots. We show that this invariant separates Seifert fibered integral homology spaces and can be used to detect the unknot.

In 2002, L. Nicolaescu and the fourth author formulated a very general conjecture which relates the geometric genus of a Gorenstein surface singularity with rational homology sphere link with the Seiberg--Witten invariant (or one of its candidates) of the link. Recently, the last three authors found some counterexamples using superisolated singularities. The theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves. In the case when the corresponding curve has only one singular point one knows no counterexample. In fact, in this case the above Seiberg--Witten conjecture led us to a very interesting and deep set of `compatibility properties' of these curves (generalising the Seiberg--Witten invariant conjecture, but sitting deeply in algebraic geometry) which seems to generalise some other famous conjectures and properties as well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yau inequalities). Namely, we provide a set of `compatibility conditions' which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer $d$ if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree $d$. The conjectured compatibility properties have a weaker version too, valid for any rational cuspidal curve with more than one singular point. The goal of the present article is to formulate these conjectured properties, and to verify them in all the situations when the logarithmic Kodaira dimension of the complement of the corresponding plane curves is strictly less than 2.

It is shown that every strongly-cyclic branched covering of a (1,1)-knot is a Dunwoody manifold. This result, together with the converse statement previously obtained by Grasselli and Mulazzani, proves that the class of Dunwoody manifolds coincides with the class of strongly-cyclic branched coverings of (1,1)-knots. As a consequence, a parametrization of (1,1)-knots by 4-tuples of integers is obtained. Moreover, using a representation of (1,1)-knots by the mapping class group of the twice-punctured torus, an algorithm is provided which gives the parametrization of all torus knots in $\mathbf{S}^3$.

A previous conjecture is verified for any normal surface singularity which admits a good ${\mathbb C}^*$-action. This result connects the Seiberg–Witten invariant of the link (associated with a certain ‘canonical’ spin$^c$ structure) with the geometric genus of the singularity, provided that the link is a rational homology sphere.

As an application, a topological interpretation is found of the generalized Batyrev stringy invariant (in the sense of Veys) associated with such a singularity.

The result is partly based on the computation of the Reidemeister–Turaev sign-refined torsion and the Seiberg–Witten invariant (associated with any spin$^c$ structure) of a Seifert 3-manifold with negative orbifold Euler number and genus zero.