A concordance invariant from the Floer homology of double branched covers

By Ciprian Manolescu, Brendan Owens. 2005

112 citations recorded by the source corpus.

Ozsvath and Szabo defined an analog of the Froyshov invariant in the form of a correction term for the grading in Heegaard Floer homology. Applying this to the double cover of the 3-sphere branched over a knot K, we obtain an invariant δ of knot concordance. We show that δ is determined by the signature for alternating knots and knots with up to nine crossings, and conjecture a similar relation for all H-thin knots. We also use δ to prove that for all knots K with τ(K) > 0, the positive untwisted double of K is not smoothly slice.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/7396d38ff2b6f8f651505a65602a08086b49d143

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Citations in the corpus

The reduced knot Floer complex (2013, methodology): “These invariants have been used to answer questions related to Dehn surgery [5, 6, 19], the smooth knot concordance group [4, 11, 15, 18] and various notions of genera of knots [1, 7, 20].”
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] (”
Pin(2)-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture (2013, methodology): “This was done for the Ozsváth-Szabó correction term d in [38].”
Topologically slice knots of smooth concordance order two (2012, methodology): “Techniques derived from Heegaard Floer theory and Khovanov homology (specifically the Rasmussen invariant [39]) were used to show that CTS contains a summand isomorphic to Z(3) [23, 24, 25].”
Heegaard Floer homology and genus one, one‐boundary component open books (2008, background): “Manolescu and Owens prove in [23] that the assignment δ(K) = 2d(Σ(K), s0) is a surjective group homomorphism δ : C → Z, where s0 is the unique self-conjugate Spin c structure on Σ(K).”
A survey on Heegaard Floer homology and concordance (2015, methodology): “k; see [HKL12]. Alternatively, if Σpn(K) can be described as the boundary of a negative deﬁnite plumbing, then the results of [OS03c] and [N´em05] (see also [CK14]) can be used to compute δpn(K). See [MO07] and [Jab12] for examples of applications of these techniques. 3.4. Ozsvath-Stipsicz-Szab´o’s ”
Sums of lens spaces bounding rational balls (2007, result): “nce group C1. The survey paper [8] describes what was known about the group C1 until 2004. More recent papers contain results on knot concordance obtained via Ozsvath–Szab´o’s Heegaard Floer homology [4, 5, 9]. In spite of all the eﬀorts made so far, the structure of the group C1 remains quite myste”
Unoriented knot Floer homology and the unoriented four-ball genus (2015, methodology): “• If p and q are odd and relatively prime, the branched double cover of Tp,q is the Brieskorn sphere Σ(2, p, q); moreover, if q = 2pn ± 1 for some integer n, then Σ(2, p, 2pn ± 1) = S3 ±1(T2,p); and hence (using the formulae from [12]) δ(Tp,2pn+1) = 0 δ(Tp,2pn−1) = −2⌈ n 2 ⌉.”
An infinite-rank summand of topologically slice knots (2013, methodology): “Building upon the techniques of Ozsváth and Szabó [29], Khovanov [22] and Rasmussen [34], Livingston’s Z–summand result can be improved to a Z(3) –summand (see Livingston [27] and Manolescu and Owens [28]), using the smooth concordance homomorphisms given by the Rasmussen s invariant, and the Manole”
Concordance homomorphisms from knot Floer homology (2014, methodology): “Similarly-behaved invariants have been constructed using more recently-defined invariants, such as knot Floer homology (the invariant τ(K) [25, 34]), Khovanov homology (Rasmussen’s s invariant [36]), and the absolute grading on Heegaard Floer homology of the double branched cover of a knot (see [19]”
Localization for Involutions in Floer Cohomology (2010, result): “The set of rational numbers encountered as differences could potentially be a knot invariant (and if one restricts to the components corresponding to Spin rather than Spinc structures, even a knot concordance invariant, following a similar philosophy to that of [15]).”
Topologically slice knots with nontrivial Alexander polynomial (2010, background): “This invariant, defined by Ozsváth and Szabó in [43], has been applied to study knot concordance in [23, 25, 34]).”
Involutive Heegaard Floer homology (2015, background): “However, the non-sliceness of 41 cannot be detected by most of the modern concordance invariants coming from Floer or Khovanov homology: τ from [37, 48], s from [51], δ from [25], d(S3 1(K)) = −2V0(K) from [36, 49, 46], ν from [45], ν+ from [9], ε from [8], and ΥK(t) from [33] all vanish on amphichi”