The knot Floer complex and the smooth concordance group

By Jennifer Hom. 2011

93 citations recorded by the source corpus.

We define a new smooth concordance homomorphism based on the knot Floer complex and an associated concordance invariant, epsilon. As an application, we show that an infinite family of topologically slice knots are independent in the smooth concordance group.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a5ab2704b36f4b85ff60a1db13ace48a35b678ca

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Citations in the corpus

Bordered Heegaard Floer homology and the tau‐invariant of cable knots (2012, methodology): “The various types of behavior of ε(K#nK′), n ∈ Z, are studied further in [2, 7], with applications to understanding the knot concordance group.”
Topologically slice knots of smooth concordance order two (2012, methodology): “Using techniques developed by Donaldson [4] and later enhanced by Fintushel-Stern [6] and Furuta [10], Endo [5] proved that CTS contains a subgroup isomorphic to Z∞ (see also [15, 16, 17] for other constructions of infinite rank free subgroups).”
More concordance homomorphisms from knot Floer homology (2019, methodology): “Theorem 2.5 ([Zem17a, Theorem 1.5], cf. [Hom14, Theorem 2]).”
Four-ball genus bounds and a refinement of the Ozsvath-Szabo tau-invariant (2014, background): “The torus knot T2,9 is an L-space knot, as is T2,3;2,5 [Hed09, Theorem 1.10], so their knot Floer complexes are completely determined by their Alexander polynomials [OS05, Theorem 1.2] (cf. [Hom11, Remark 6.6]).”
On the geography and botany of knot Floer homology (2014, background): “Here, two such complexes C, D are equivalent if (C ⊗ D∗) = 0 (with D∗ the dual complex), with the product and inverse operations in CFK given by tensor product and duality, respectively [34,35].”
AN INTRODUCTION TO KNOT FLOER HOMOLOGY (2014, unknown): “By using the full knot Floer complex, one can extract additional concordance invariants; see [Hom14b, Hom15, HW14].”
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”
A survey on Heegaard Floer homology and concordance (2015, methodology): “…Heegaard Floer theory: • Ozsváth-Szabó’s Z-valued invariant τ [OS03b], • Ozsváth-Szabó’s Z-valued invariant ν [OS11], • the author’s {−1, 0, 1}-valued invariant ε [Hom14b], • the author and Wu’s ν+ (equivalently, ν−) invariant [HW16], based on work of Rasmussen [Ras03], • the sequence of…”
A connected sum formula for involutive Heegaard Floer homology (2016, unknown): “The construction of I is reminiscent to that of the group of knot Floer complexes modulo ε-equivalence, defined by Hom in [7].”
An infinite-rank summand of topologically slice knots (2013, methodology): “Furthermore, the group CFK is totally ordered [Hom11a, Proposition 4.1], with the ordering given by C1 > C2 if and only if ε(C1 ⊗ C ∗ 2 ) = 1.”
Connected sums and involutive knot Floer homology (2017, methodology): “h concordance group. Using Theorem1.1we construct a homomorphism from the smooth concordance group Cto an algebraically dened, abelian group I K. The construction is inspired by constructions of Hom [Hom14] and Storegen [Sto15]. See also [HMZ18] for a similar construction. In Section2, we dene a not”