Knot Floer homology and the four-ball genus

By Peter S. Ozsv'ath, Zolt'an Szab'o. 2003

498 citations recorded by the source corpus.

We use the knot filtration on the Heegaard Floer complex to define an integer invariant tau(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, tau gives sharp bounds on the four-ball genera of torus knots. As another illustration, we calculate the invariant for several ten-crossing knots.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4d26ed75ca9c62908245de53b086600bc5a67f8b

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Citations in the corpus

An infinite-rank summand of topologically slice knots (2013, methodology): “The first result in this direction is due to Livingston [26], who, using the Ozsváth–Szabó invariant [30], showed that CTS contains a summand isomorphic to Z and thus contains elements that are not divisible.”
Khovanov homology and the slice genus (2004, methodology): “Combining the results of [4], [11], [13], [14], and [15] with some unpublished computations of the author, it appears that the value of τ has been determined for all but two knots of 10 crossings and fewer.”
A Survey of Classical Knot Concordance (2003, methodology): “The other important development concerns the application of differential geometric techniques to the study of smooth 4–manifolds, beginning with the work of Donaldson [20] [21] and including the introduction of Seiberg–Witten invariants and their application to symplectic manifolds, the use of the T”
On knot Floer homology and lens space surgeries (2003, background): “3 of [20]) and also [23], it is shown that if g(K) denotes the four-ball genus, then”
The Ozsváth-Szabó and Rasmussen Concordance Invariants are not Equal (2005, background): “´ O AND RASMUSSEN CONCORDANCE INVARIANTS ARE NOT EQUAL 11´ Next, recall that τ (and s) satisfy the following inequality under the operation of changing a crossing in a given knot diagram (see [14] or [22] for a proof): τ(K+)− 1 ≤ τ(K−) ≤ τ(K+), where K+ (resp. K−) denote the diagram with the positiv”
NOTIONS OF POSITIVITY AND THE OZSVÁTH–SZABÓ CONCORDANCE INVARIANT (2005, background): “In [21] Ozsváth and Szabó introduced an integer-valued invariant τ(K) associated to a knot, K ⊂ S(3) (see also [29].”
Knot Floer homology and rational surgeries (2005, unknown): “3 of [16] and Proposition 3.”
Involutive Heegaard Floer homology (2015, background): “The value τ = τ(K) is the knot concordance invariant defined in [48, 37].”
On Knot Floer Homology and Cabling: 2 (2008, background): “2There will be no loss of generality in doing this, since K −p,−q ≃ −Kp,q where −Kp,q is Kp,q with reversed string orientation.”
Transverse knots and Khovanov homology (2004, methodology): “with the s(K) invariant of Bar-Natan [BN], and is twice the τ (K) invariant of Ozsv´ ath and Szab´ o [ OS1 ].”
Unoriented knot Floer homology and the unoriented four-ball genus (2015, result): “The genus bounds obtained here are similar to earlier results; for example, those of [1] and [10] in the non-orientable case and [19] and [25] in the orientable case.”
Computations of Heegaard-Floer knot homology (2006, methodology): “This is the well-known concordance invariant defined by Ozsváth and Szabó in [20].”
Floer homology and knot complements (2003, background): “W e say thatK is perfectifthe hom ologicaland Alexander gradings on dC F r(K )are the sam e. IfK is perfect,the rank of dC F (j) r (K ) is equalto the absolute value ofthe coe cient oftj in K (t).”
Filtering smooth concordance classes of topologically slice knots (2012, background): “Here τ is the concordance invariant of Ozsváth-Szabó and Rasmussen defined from Heegard Floer homology [45], is the concordance invariant of Hom [35], s is Rasmussen’s concordance invariant defined from Khovanov homology [49], and δp refers to the invariants of Manolescu-Owens [41] and Jabuka [37]”