Symmetries and Adjunction Inequalities for Knot Floer Homology

Abstract

We derive symmetries and adjunction inequalities of the knot Floer homology groups which appear to be especially interesting for homologically essential knots. Furthermore, we obtain an adjunction inequality for cobordism maps in knot Floer homologies. We demonstrate the adjunction inequalities and symmetries in explicit calculations which recover some of the main results from [E. Eftekhary, Longitude Floer homology and the Whitehead double, Algebr. Geom. Topol. 5 (2005) 1389-1418] on longitude Floer homology and also give rise to vanishing results on knot Floer homologies.of K. In the case [K] = 0 this filtration gets lost. The information given by the filtration, however, does not seem to get lost (at least not fully), but are shifted into the Spin c -refinements of the groups. We find it natural to also study the groups for [K] = 0. The first step in this study is to provide tools making the homology groups accessible to computations.Recall that Heegaard Floer homology groups can be computed by applying the following tools: Surgery exact triangles, adjunction inequalities of Heegaard Floer groups and adjunction inequalities of maps induced by cobordisms. The groups HFK may be equipped with an adjunction inequality coming from sutured Floer homology (see [6, Theorem 2]). Furthermore, Juhász's work on cobordism maps for sutured Floer homologies provide a notion of cobordism maps for the HFK-case, as well (see [5]). In [16], we introduced cobordism maps for various versions of knot Floer homology, in particular for HFK •,• (i.e. HFK •,• = HFK, HFK •,− and HFK •,∞ , see Sec. 2.2 for a definition).In this paper, we study symmetry properties of knot Floer homology groups HFK •,• , adjunction inequalities for HFK •,• and adjunction inequalities for cobordism maps of these groups. After discussing these concepts, we present some implications of these results which are meant as a demonstration how these techniques may be applied in computations. We have to point out that the computational results we present (for the HFK-homology) can be alternatively derived (and some strengthened) from work of Friedl, Juhász and Rasmussen on sutured Floer homology (see [2, Proposition 7.7]).

On concordances in 3-manifolds

“…The same holds for rationally nullhomologous knots, and similar results hold also in the general case, see .…”