We derive a new exact sequence in the hat-version of Heegaard Floer homology. As a consequence we see a functorial connection between the invariant of Legendrian knots L and the contact element. As an application we derive two vanishing results of the contact element making it possible to easily read off its vanishing out of a surgery presentation in suitable situations. 57R17; 53D35, 57R58
Abstract. We apply spectral sequences to derive both an obstruction to the existence of n-fold prolongations and a topological classification. Prolongations have been used in the literature in an attempt to prove that every Engel structure on M × S 1 with characteristic line field tangent to the fibers is determined by the contact structure induced on a cross section and the twisting of the Engel structure along the fibers. Our results show that this statement needs some modification: to classify the diffeomorphism type of the Engel structure we additionally have to fix a class in the first cohomology of M .
No 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]).
This paper provides an introduction to the basics of Heegaard Floer homology with some emphasis on the hat theory and to the contact geometric invariants in the theory. The exposition is designed to be comprehensible to people without any prior knowledge of the subject.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.