Floer homology and knot complements

By Jacob Rasmussen. 2003

688 citations recorded by the source corpus.

We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call CF_r. It carries information about the Floer homology of large integral surgeries on the knot. Using the exact triangle, we derive information about other surgeries on knots, and about the maps on Floer homology induced by certain surgery cobordisms. We define a certain class of \em{perfect} knots in S^3 for which CF_r has a particularly simple form. For these knots, formal properties of the Ozsvath-Szabo theory enable us to make a complete calculation of the Floer homology. This is the author's thesis; many of the results have been independently discovered by Ozsvath and Szabo in math.GT/0209056.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/c1ded9d368cff8d7cbc64c5e66989753207811fe

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Citations in the corpus

Heegard Floer invariants of Legendrian knots in contact three-manifolds (2008, positive): “The Heegaard Floer homology groups HF(Y ), ĤF(Y ) of a 3-manifold were introduced in [33] and extended in the case where Y is equipped with a null-homologous knotK ⊂ Y to variants HFK(Y,K), ĤFK(Y,K) in [35, 40].”
The decategorification of sutured Floer homology (2009, positive): “It is an offshoot of the Heegaard Floer homology of Ozsváth and Szabó [26], and contains knot Floer homology [28, 33] as a special case.”
On the geography and botany of knot Floer homology (2014, background): “Jake Rasmussen also observed that the second highest group of a knot admitting L-space surgeries is non-trivial, as well as the utility of his inequality for the h-invariant, leading to the proof of Theorem 7.”
Grid Homology for Knots and Links (2015, methodology): “Intriguing though these similarities are, a precise mathematical relationship between the constructions of knot Floer and Khovanov homologies has yet to be discovered; compare [36, 191].”
Cables of thin knots and bordered Heegaard Floer homology (2009, background): “S(3)/, Ozsváth and Szabó define an integer knot invariant [14], independently discovered by Rasmussen [18], whose absolute value is a lower bound on the four-ball genus.”
Connected sums and involutive knot Floer homology (2017, background): “ive Heegaard Floer homology. Knot Floer homology is a renement of Heegaard Floer homology for knots embedded in 3- manifolds, introduced by Ozsvath and Szabo [OS04a] and independently by Rasmussen [Ras03]. Link Floer homology is a generalization of knot Floer homology for links in 3-manifolds, devel”
A combinatorial spanning tree model for knot Floer homology (2011, methodology): “Knot Floer homology is an invariant of oriented links in the 3-sphere, originally defined by Ozsváth-Szabó [36] and by Rasmussen [44] using Heegaard diagrams and holomorphic disks.”
A survey of Heegaard Floer homology (2013, background): “Note that Rasmussen introduced the s-invariant using Khovanov homology, which also gives a lower bound on the 4-ball genus, as |spKq| ď 2g˚pKq. Rasmussen conjectured that spKq “ 2τpKq, which was then disproved by Hedden and Ording [29].”
Matrix factorizations and link homology II (2005, methodology): “and R1 x 3−x6”
Cobordisms of sutured manifolds and the functoriality of link Floer homology (2009, background): “s. Finally, we show that a Weinstein sutured manifold cobordism preserves the contact element. 1. Introduction Knot Floer homology, introduced independently by Ozsv´ath and Szabo´ [35], and Rasmussen [41], and later generalized to links by Ozsv´ath and Szab´o [39], has proven to be a very sensitive ”
Bordered Heegaard Floer homology and the tau‐invariant of cable knots (2012, background): “To a knot K ⊂ S(3), Ozsváth and Szabó [13], and independently Rasmussen [19], associate a Z⊕ Z-filtered chain complex CFK∞(K), whose doubly filtered chain homotopy type is an invariant of K.”
Computations of Heegaard-Floer knot homology (2006, background): “[26] J. Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University, 2003.”
On Knot Floer Homology and Cabling: 2 (2008, methodology): “Does the Rasmussen concordance invariant, applied to all iterated cables of K, determine if K is smoothly slice?”
Spanning trees and Khovanov homology (2006, background): “ning tree complex C(D) that makes it chain homotopic to Ce(D): Theorem 6. For coeﬃcients in a ﬁeld F, there exists a diﬀerential on C(D) such that He ∗, (D;F) ∼= H∗ ∗ (C(D);F). Proof: By Lemma 4.5 in [7], there exists a unique ﬁltered complex C′ which is chain homotopic to Ce(D) and C′ ∼= H∗(F pCe(D”
On combinatorial link Floer homology (2006, methodology): “G T ] 2 6 M ar 2 01 0”
Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology (2007, result): “f knots in lens spaces, making no mention of J–holomorphic curves. We postpone the proof that our chain complex is isomorphic to the one deﬁned by Ozsv´ath and Szabo´ in [OS05a] (see also [OS04b] and [Ras03]) until Section 4, after having reviewed the relevant aspects of Heegaard Floer theory in Sec”
A cylindrical reformulation of Heegaard Floer homology (2005, background): “Since its discovery around the turn of the millennium, Heegaard Floer homology has been applied by Ozsváth, Rasmussen and Szabó to the study of knots and surgery [19, 25, 18], contact structures [23] and symplectic structures [22], and is strong enough to reprove most results about smooth four–manif”
Holomorphic disks and knot invariants (2002, background): “The map φ is said to be homogeneous if it is homogeneous of degree d, for some d.”