Connected Floer homology of covering involutions
4
Citation
35
Reference
20
Related Paper
Citation Trend
Abstract:
Using the covering involution on the double branched cover of the three-sphere branched along a knot, and adapting ideas of Hendricks-Manolescu and Hendricks-Hom-Lidman, we define new knot invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots.Keywords:
Floer Homology
Covering space
Cite
Journal Article A Concordance Invariant from the Floer Homology of Double Branched Covers Get access Ciprian Manolescu, Ciprian Manolescu 1 Department of Mathematics, Columbia University New York, NY 10027, USA Correspondence to be sent to: Ciprian Manolescu, Department of Mathematics, Columbia University New York, NY 10027, USA. e-mail: cm@math.columbia.edu Search for other works by this author on: Oxford Academic Google Scholar Brendan Owens Brendan Owens 2 Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2007, 2007, rnm077, https://doi.org/10.1093/imrn/rnm077 Published: 01 January 2007 Article history Received: 27 September 2006 Revision received: 27 September 2006 Published: 01 January 2007 Accepted: 30 July 2007
Floer Homology
Khovanov homology
Concordance
Homology
Cite
Citations (102)
Using the conjugation symmetry on Heegaard Floer complexes, we define a 3-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to Z4-equivariant Seiberg–Witten Floer homology. Further, we obtain two new invariants of homology cobordism, d̲ and d¯, and two invariants of smooth knot concordance, V̲0 and V¯0. We also develop a formula for the involutive Heegaard Floer homology of large integral surgeries on knots. We give explicit calculations in the case of L-space knots and thin knots. In particular, we show that V̲0 detects the nonsliceness of the figure-eight knot. Other applications include constraints on which large surgeries on alternating knots can be homology-cobordant to other large surgeries on alternating knots.
Floer Homology
Morse homology
Heegaard splitting
Cobordism
Khovanov homology
Homology
Cite
Citations (106)
Knot Floer homology is an invariant for knots in the three-sphere for which the Euler characteristic is the Alexander–Conway polynomial of the knot. The aim of this paper is to study this homology for a class of satellite knots, so as to see how a certain relation between the Alexander–Conway polynomials of the satellite, companion and pattern is generalized on the level of the knot Floer homology. We also use our observations to study a classical geometric invariant, the Seifert genus, of our satellite knots.
Floer Homology
Khovanov homology
Knot polynomial
Alexander polynomial
Homology
Tricolorability
Cite
Citations (0)
We use monopole Floer homology for sutured manifolds to construct invariants of Legendrian knots in a contact 3-manifold.These invariants assign to a knot K ⊂ Y elements of the monopole knot homology KHM (-Y, K), and they strongly resemble the knot Floer homology invariants of Lisca, Ozsváth, Stipsicz, and Szabó.We prove several vanishing results, investigate their behavior under contact surgeries, and use this to construct many examples of non-loose knots in overtwisted 3manifolds.We also show that these invariants are functorial with respect to Lagrangian concordance.Throughout this paper we will adopt the convention that letters in the standard math font, such as K, refer to topological knots, whereas the same letters in a script font, such as K, refer to Legendrian representatives of those knot types.We also remark that Lekili [29] has shown that one can replace HM with HF + in the Kronheimer-Mrowka construction of sutured monopole homology in order to recover sutured Floer homology.Thus the reader can apply the constructions in this paper to obtain a similar Legendrian invariant in knot Floer homology, and everything in this paper will still hold except the Lagrangian concordance results of section 6.In this sense we conjecture that ℓ(K) is identical to the LOSS invariant.
Floer Homology
Homology
Khovanov homology
Morse homology
Cite
Citations (16)
Author(s): Zemke, Ian Michael | Advisor(s): Manolescu, Ciprian | Abstract: In the early 2000s, Ozsv #x27;{a}th and Szab #x27;{o} introduced a collection of invariants for 3--manifolds and 4--manifolds called Heegaard Floer homology. To a 3--manifold they constructed a group, and to a 4--manifold which cobounds two 3--manifolds, they constructed a homomorphism between the manifolds appearing on the ends. Their invariants satisfy many of the axioms of a TQFT as described by Atiyah, however their construction has some additional restrictions which prevent it from fitting into Atiyah's framework. There is a refinement of Heegaard Floer homology for 3--manifolds containing a knot, due to Ozsv #x27;{a}th and Szab #x27;{o}, and independently Rasmussen, and a further refinement for 3--manifolds containing links, due to Ozsv #x27;{a}th and Szab #x27;{o}. It's a natural question as to whether one can define functorial maps associated to link cobordisms.In this thesis, we describe a package of cobordism maps for Heegaard Floer homology and link Floer homology. The cobordism maps satisfy an appropriate analogy of the axiomatic description of a TQFT formulated by Atiyah. To a ribbon graph cobordism between two based 3--manifolds, we associate a map between the Heegaard Floer homologies of the ends. To a decorated link cobordism, we obtain maps on the link Floer homologies of the ends. The maps associated to decorated link cobordisms reduce to the maps for ribbon graphs, in a natural way. As applications, we describe several formulas for mapping class group actions on the Heegaard Floer and knot Floer groups. We prove a new bound on a concordance invariant $\Upsilon_K(t)$ from knot Floer homology, and also see how the link cobordism maps give straightforward proofs of other bounds on concordance invariants from knot Floer homology. We also explore the interaction of the maps with conjugation actions on Heegaard Floer homology and link Floer homology, giving connected sum formulas for involutive Heegaard Floer homology and involutive knot Floer homology.
Cobordism
Floer Homology
Heegaard splitting
Homology
Ribbon
Cite
Citations (0)
This paper explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist bigraded groups satisfying all previously known constraints of knot Floer homology which do not arise as the invariant of a knot. This leads to a new constraint for knots admitting lens space surgeries, as well as a proof that the rank of knot Floer homology detects the trefoil knot. For the second, we show that any non-trivial band sum of two unknots gives rise to an infinite family of distinct knots with isomorphic knot Floer homology. We also prove that the fibered knot with identity monodromy is strongly detected by its knot Floer homology, implying that Floer homology solves the word problem for mapping class groups of surfaces with non-empty boundary. Finally, we survey some conjectures and questions and, based on the results described above, formulate some new ones.
Floer Homology
Knot complement
Cite
Citations (63)
In an earlier work, we introduced a family of t-modified knot Floer homologies, defined by modifying the construction of knot Floer homology HFK-minus. The resulting groups were then used to define concordance homomorphisms indexed by t in [0,2]. In the present work we elaborate on the special case t=1, and call the corresponding modified knot Floer homology the unoriented knot Floer homology. Using elementary methods (based on grid diagrams and normal forms for surface cobordisms), we show that the resulting concordance homomorphism gives a lower bound for the smooth 4-dimensional crosscap number of a knot K --- the minimal first Betti number of a smooth (possibly non-orientable) surface in the 4-disk that meets the boundary 3-sphere along the given knot K.
Floer Homology
Ball (mathematics)
Homology
Cite
Citations (31)
We prove that the knot Floer homology of a fibered knot is nontrivial in its next-to-top Alexander grading. Immediate applications include new proofs of Krcatovich's result that knots with $L$-space surgeries are prime and Hedden and Watson's result that the rank of knot Floer homology detects the trefoil among knots in the 3--sphere. We also generalize the latter result, proving a similar theorem for nullhomologous knots in any 3--manifold. We note that our method of proof inspired Baldwin and Sivek's recent proof that Khovanov homology detects the trefoils. As part of this work, we also introduce a numerical refinement of the Ozsv\'ath-Szab\'o contact invariant. This refinement was the inspiration for Hubbard and Saltz's annular refinement of Plamenevskaya's transverse link invariant in Khovanov homology.
Floer Homology
Khovanov homology
Knot complement
Knot polynomial
Homology
Quantum invariant
Cite
Citations (17)
Let K be a rationally null-homologous knot in a three-manifold Y .We construct a version of knot Floer homology in this context, including a description of the Floer homology of a three-manifold obtained as Morse surgery on the knot K .As an application, we express the Heegaard Floer homology of rational surgeries on Y along a null-homologous knot K in terms of the filtered homotopy type of the knot invariant for K .This has applications to Dehn surgery problems for knots in S 3 .In a different direction, we use the techniques developed here to calculate the Heegaard Floer homology of an arbitrary Seifert fibered three-manifold with even first Betti number.
Floer Homology
Morse homology
Homology
Cite
Citations (234)
Abstract Using the covering involution on the double branched cover of $$S^3$$ S 3 branched along a knot, and adapting ideas of Hendricks–Manolescu and Hendricks–Hom–Lidman, we define new knot (concordance) invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots.
Floer Homology
Concordance
Merge (version control)
Cite
Citations (14)