Abstract. We give bounds on the gap functions of the singularities of a cuspidal plane curve of arbitrary genus, generalising recent work of Borodzik and Livingston. We apply these inequalities to unicuspidal curves whose singularity has one Puiseux pair: we prove two identities tying the parameters of the singularity, the genus, and the degree of the curve; we improve on some degree-multiplicity asymptotic inequalities; finally, we prove some finiteness results, we construct infinite families of examples, and in some cases we give an almost complete classification.
In this note we use Heegaard Floer homology to study smooth cobordisms of algebraic knots and complex deformations of cusp singularities of curves. The main tool will be the concordance invariant ν + : we study its behaviour with respect to connected sums, providing an explicit formula in the case of L-space knots and proving subadditivity in general.
We describe an action of the concordance group of knots in S3 on concordances of knots in arbitrary 3‐manifolds. As an application we define the notion of almost‐concordance between knots. After some basic results, we prove the existence of non‐trivial almost‐concordance classes in all non‐abelian 3‐manifolds. Afterwards, we focus the attention on the case of lens spaces, and use a modified version of the Ozsváth–Szabó–Rasmussen's τ‐invariant to obstruct almost‐concordances and prove that each L(p,1) admits infinitely many nullhomologous non almost‐concordant knots. Finally we prove an inequality involving the cobordism PL‐genus of a knot and its τ‐invariants, in the spirit of [Sarkar, Math. Res. Lett. 18 (2011) 1239–1257].
We consider the question of when a rational homology
$3$
-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology group injects in the first homology group of any other element in the same class. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite-rank cokernels. Further consequences include a divisibility condition between the determinants of a connected sum of
$2$
-bridge knots and any other knot in the same concordance class. Lastly, we use knot Floer homology combined with our main result to obstruct Dehn surgeries on knots from being rationally cobordant to lens spaces.
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