Stanislav Jabuka

The slice-ribbon conjecture for 3-stranded pretzel knots

We determine the (smooth) concordance order of the 3-stranded pretzel knots

Product formulae for Ozsváth-Szabó 4-manifold invariants

P(p, q, r)

Heegaard Floer homology of certain mapping tori

with

WHEN ARE TWO DEDEKIND SUMS EQUAL

p, q, r

Periodic knots and Heegaard Floer correction terms

odd. We show that each one of finite order is, in fact, ribbon, thereby proving the slice-ribbon conjecture for this family of knots. As corollaries we give new proofs of results first obtained by Fintushel-Stern and Casson-Gordon.

The nonorientable 4–genus for knots with 8 or 9 crossings

We give formulae for the Ozsvath‐Szabo invariants of 4‐manifolds X obtained by fiber sum of two manifolds M1 , M2 along surfaces U1 , U2 having trivial normal bundle and genus g 1. The formulae follow from a general theorem on the Ozsvath‐ Szabo invariants of the result of gluing two 4‐manifolds along a common boundary, which is phrased in terms of relative invariants of the pieces. These relative invariants take values in a version of Heegaard Floer homology with coefficients in modules over certain Novikov rings; the fiber sum formula follows from the theorem that this “perturbed” version of Heegaard Floer theory recovers the usual Ozsvath‐Szabo invariants, when the 4‐manifold in question has b C 2. The construction allows an extension of the definition of Ozsvath‐Szabo invariants to 4‐manifolds having b C D 1 depending on certain choices, in close analogy with Seiberg‐Witten theory. The product formulae lead quickly to calculations of the Ozsvath‐Szabo invariants of various 4‐manifolds; in all cases the results are in accord with the conjectured equivalence between Ozsvath‐Szabo and Seiberg‐Witten invariants.

Order in the concordance group and Heegaard Floer homology

We calculate the Heegaard Floer homologies HF + (M, s) for mapping tori M associated to certain surface diffeomorphisms, where s is any Spin c structure on M whose first Chern class is non-torsion. Let γ and δ be a pair of geometrically dual nonseparating curves on a genus g Riemann surfaceg, and let σ be a curve separatingg into components of genus 1 and g −1. Write t, t�, and tfor the right-handed Dehn twists about each of these curves. The examples we consider are the mapping tori of the diffeomorphisms t m ◦ t n for m, n ∈ Z and that of t ±1 � . AMS Classification 57R58; 53D40

On the Heegaard Floer homology of a surface times a circle

A natural question about Dedekind sums is to find conditions on the integers

Concordance invariants from higher order covers

a_1, a_2

Heegaard Floer Correction Terms and Dedekind–Rademacher Sums

, and