Dehn twists in Heegaard Floer homology

Sahamie, Bijan

doi:10.2140/agt.2010.10.465

Cited by 13 publications

(29 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This computational ease underpins many of the known results regarding families of transversely non-simple knot types: Ng, Ozsváth and Thurston [26], Vértesi [41], Baldwin [3], and Khandhawit and Ng [19]. On the other hand, the more geometric construction in [21] makes it possible to establish general properties of the LOSS invariants that are hard to prove for the GRID invariants, like Ozsváth and Stipsicz's result on their behaviors under .C1/-contact surgeries [29], or Sahamie's result relating ᏸ.L/ to the contact invariant of .C1/-contact surgery on L [38]. Further, it is conjectured that the Legendrian invariants defined in [33] and [21] are well-behaved with respect to Lagrangian cobordism.…”

Section: Introductionmentioning

confidence: 99%

On the equivalence of Legendrian and transverse invariants in knot Floer homology

2013

View full text Add to dashboard Cite

Using the grid diagram formulation of knot Floer homology, Ozsváth, Szabó and Thurston defined an invariant of transverse knots in the tight contact 3-sphere. Shortly afterwards, Lisca, Ozsváth, Stipsicz and Szabó defined an invariant of transverse knots in arbitrary contact 3-manifolds using open book decompositions. It has been conjectured that these invariants agree where they are both defined. We prove this fact by defining yet another invariant of transverse knots, showing that this third invariant agrees with the two mentioned above.

show abstract

Section: Introductionmentioning

confidence: 99%

On the equivalence of Legendrian and transverse invariants in knot Floer homology

2013

View full text Add to dashboard Cite

show abstract

“…Knot Floer homologies have also proven to be very useful in knot theoretic applications, the filtration on these groups carrying a lot of geometric information. In [17], we made the observation that the knot Floer homology is not restricted to homologically trivial knots. For [K] = 0 the knot theoretic information was especially encoded into a filtration constructed using a Seifert surface of K. In the case [K] = 0 this filtration gets lost.…”

Section: Introductionmentioning

confidence: 99%

Symmetries and Adjunction Inequalities for Knot Floer Homology

Sahamie¹

2012

J. Knot Theory Ramifications

Self Cite

View full text Add to dashboard Cite

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]).

show abstract

“…Additional properties of L. Here, we describe some additional properties satisfied by L which are analogous to those satisfied by the LOSS invariant [31,39,36] and the second author's invariant ℓ from [40]. We will omit basepoints from our notation for convenience and because they are not so relevant to the results.…”

Section: 3mentioning

confidence: 99%

Invariants of Legendrian and transverse knots in monopole knot homology

Baldwin

Sivek

2018

Journal of Symplectic Geometry

View full text Add to dashboard Cite

We use the contact invariant defined in [2] to construct a new invariant of Legendrian knots in Kronheimer and Mrowka's monopole knot homology theory (KHM ), following a prescription of Stipsicz and Vértesi. Our Legendrian invariant improves upon an earlier Legendrian invariant in KHM defined by the second author in several important respects. Most notably, ours is preserved by negative stabilization. This fact enables us to define a transverse knot invariant in KHM via Legendrian approximation. It also makes our invariant a more likely candidate for the monopole Floer analogue of the "LOSS" invariant in knot Floer homology. Like its predecessor, our Legendrian invariant behaves functorially with respect to Lagrangian concordance. We show how this fact can be used to compute our invariant in several examples. As a byproduct of our investigations, we provide the first infinite family of nonreversible Lagrangian concordances between prime knots.

show abstract

Dehn twists in Heegaard Floer homology

Cited by 13 publications

References 25 publications

On the equivalence of Legendrian and transverse invariants in knot Floer homology

On the equivalence of Legendrian and transverse invariants in knot Floer homology

Symmetries and Adjunction Inequalities for Knot Floer Homology

Invariants of Legendrian and transverse knots in monopole knot homology

Contact Info

Product

Resources

About