Ana G. Lecuona

On the slice-ribbon conjecture for Montesinos knots

We establish the slice-ribbon conjecture for a family P of Montesinos knots by means of Donaldson’s theorem on the intersection forms of definite 4-manifolds. The 4-manifolds that we consider are obtained by plumbing disc bundles over S2 according to a star-shaped negative-weighted graph with 3 legs such that: i) the central vertex has weight less than or equal to −3; ii) − total weight − 3 #vertices < −1. The Seifert spaces which bound these 4-dimensional plumbing manifolds are the double covers of S3 branched along the Montesinos knots in the family P.

On the slice-ribbon conjecture for pretzel knots

We give a necessary, and in some cases sufficient, condition for sliceness inside the family of pretzel knots P.p1;:::;pn/ with one pi even. The 3‐stranded case yields two interesting families of examples: The first consists of knots for which the nonsliceness is detected by the Alexander polynomial while several modern obstructions to sliceness vanish. The second family has the property that the correction terms from Heegaard‐Floer homology of the double branched covers of these knots do not obstruct the existence of a rational homology ball; however, the Casson‐Gordon invariants show that the double branched covers do not bound rational homology balls. 57M25

Stein fillable Seifert fibered 3–manifolds

We characterize the closed, oriented, Seifert fibered 3 –manifolds which are oriented boundaries of Stein manifolds. We also show that for this class of 3 –manifolds the existence of Stein fillings is equivalent to the existence of symplectic fillings.

The Signature of a Splice

We study the behavior of the signature of colored links [Flo05, CF08] under the splice operation. We extend the construction to colored links in integral homology spheres and show that the signature is almost additive, with a correction term independent of the links. We interpret this correction term as the signature of a generalized Hopf link and give a simple closed formula to compute it.