Knot concordance and Heegaard Floer homology invariants in branched covers

Abstract

By studying the Heegaard Floer homology of the preimage of a knot K S 3 inside its double branched cover, we develop simple obstructions to K having finite order in the classical smooth concordance group. As an application, we prove that all 2-bridge knots of crossing number at most 12 for which the smooth concordance order was previously unknown have infinite smooth concordance order.57R58, 57M25; 57M12, 57M27

Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology

“…Link Floer homology in manifolds other than S 3 has been shown to detect the Thurston norm of a link complement [OS06a], [Ni06a], whether a knot is fibered [Ni06b] and has had applications to questions related to the concordance classes of knots in S 3 through the double-branched covering operation [MO05,JN06,GRS07,GJ07].…”

New constructions of slice links

“…Therefore the present paper complements these results, giving hope for an eventual complete resolution of the question of when this construction results in a slice knot or link. Our result also provides a method of producing interesting examples for testing the new obstructions to a knot or link being smoothly slice [OS03,Ra04,MO05,BW05,GRS07].…”

The concordance genus of a knot, II

“…Many new methods have been applied to obstruct torsion in C , and these have resolved many of the basic examples taken from the table of low-crossing number knots. See, for example, Grigsby, Ruberman and Strle [9], Jabuka and Naik [11], Lisca [14] and Livingston and Naik [17]. New test cases can be found by examining such algebraic concordance relations.…”