We determine the (smooth) concordance order of the 3-stranded pretzel knots P (p, q, r) with p, q, r 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

We give formulae for the Ozsváth-Szabó invariants of 4-manifolds X obtained by fiber sum of two manifolds M 1 , M 2 along surfaces † 1 , † 2 having trivial normal bundle and genus g 1. The formulae follow from a general theorem on the Ozsváth-Szabó 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 Ozsváth-Szabó invariants, when the 4-manifold in question has b 57R58; 57M99

979Order in the concordance group and Heegaard Floer homologyWe use the Heegaard-Floer homology correction terms defined by Ozsváth-Szabó to formulate a new obstruction for a knot to be of finite order in the smooth concordance group. This obstruction bears a formal resemblance to that of Casson and Gordon but is sensitive to the difference between the smooth versus topological category. As an application we obtain new lower bounds for the concordance order of small crossing knots.

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 gamma and delta be a pair of geometrically dual nonseparating curves on a genus g Riemann surface Sigma_g, and let sigma be a curve separating Sigma_g into components of genus 1 and g-1. Write t-gamma, t_delta, and t_sigma for the right-handed Dehn twists about each of these curves. The examples we consider are the mapping tori of the diffeomorphisms t_gamma^m circ t_delta^n for m,n in Z and that of t_sigma^{+-1}.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-31.abs.htm

We make a detailed study of the Heegaard Floer homology of the product of a closed surface Σg of genus g with S 1 . We determine HF + (Σg × S 1 , s; C) completely in the case c 1 (s) = 0, which for g ≥ 3 was previously unknown. We show that in this case HF ∞ is closely related to the cohomology of the total space of a certain circle bundle over the Jacobian torus of Σg, and furthermore that HF + (Σg × S 1 , s; Z) contains nontrivial 2-torsion whenever g ≥ 3 and c 1 (s) = 0. This is the first example known to the authors of torsion in Z-coefficient Heegaard Floer homology. Our methods also give new information on the action of H 1 (Σg × S 1 ) on HF + (Σg × S 1 , s) when c 1 (s) is nonzero.

We generalize the Manolescu-Owens smooth concordance invariant δ(K) of knots K ⊂ S 3 to invariants δ p n (K) obtained by considering covers of order p n , with p a prime. Our main result shows that for any prime p = 2, the thus obtained homomorphism ⊕ n∈N δ p n from the smooth concordance group to Z ∞ has infinite rank. We also show that unlike δ, these new invariants typically are not multiples of the knot signature, even for alternating knots. A significant portion of the article is devoted to exploring examples.

A natural question about Dedekind sums is to find conditions on the integers a 1 , a 2 , and b such that s(a 1 , b) = s(a 2 , b). We prove that if the former equality holds then b | (a 1 a 2 − 1)(a 1 − a 2 ). Surprisingly, to the best of our knowledge such statements have not appeared in the literature. A similar theorem is proved for the more general Dedekind-Rademacher sums as well, namely that for any fixed non-negative integer n, a positive integer modulus b, and two integers a 1 and a 2 that are relatively prime to b, the hypothesis r n (a 1 , b) = r n (a 2 , b) implies that b | (6n 2 + 1 − a 1 a 2 )(a 2 − a 1 ).2010 Mathematics Subject Classification:11F20.

The non-orientable 4-genus of a knot in the 3-sphere is defined as the smallest first Betti number of any non-orientable surface smoothly and properly embedded in the 4-ball, with boundary the given knot. We compute the non-orientable 4-genus for all knots with crossing number 8 or 9. As applications we prove a conjecture of Murakami's and Yasuhara's, and give a new lower bound for the slicing number of knot.2010 Mathematics Subject Classification. 57M25 and 57M27.