Truncated Heegaard Floer homology and knot concordance invariants

In 2003, Ozsváth and Szabó defined the concordance invariant [Formula: see text] for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of [Formula: see text] for knots in [Formula: see text] and a combinatorial proof that [Formula: see text] gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory for transverse spatial graphs in [Formula: see text], extending HFK for knots. We define a [Formula: see text]-filtered chain complex for balanced spatial graphs whose associated graded chain complex has homology determined by Harvey and O’Donnol’s graph Floer homology. We use this to show that there is a well-defined [Formula: see text] invariant for balanced spatial graphs generalizing the [Formula: see text] knot concordance invariant. In particular, this defines a [Formula: see text] invariant for links in [Formula: see text]. Using techniques similar to those of Sarkar, we show that our [Formula: see text] invariant is an obstruction to a link being slice.

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Knot concordance and Heegaard Floer homology invariants in branched covers

Geometry & Topology ◽

10.2140/gt.2008.12.2249 ◽

2008 ◽

Vol 12 (4) ◽

pp. 2249-2275 ◽

Cited By ~ 16

Author(s):

J Elisenda Grigsby ◽

Daniel Ruberman ◽

Sašo Strle

Keyword(s):

Floer Homology ◽

Heegaard Floer Homology ◽

Branched Covers ◽

Knot Concordance ◽

Heegaard Floer

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Primary decomposition in the smooth concordance group of topologically slice knots

Forum of Mathematics Sigma ◽

10.1017/fms.2021.46 ◽

2021 ◽

Vol 9 ◽

Author(s):

Jae Choon Cha

Keyword(s):

Floer Homology ◽

Heegaard Floer Homology ◽

Primary Decomposition ◽

Direct Sum ◽

Infinite Rank ◽

Knot Concordance ◽

Large Subgroup ◽

Heegaard Floer

Abstract We address primary decomposition conjectures for knot concordance groups, which predict direct sum decompositions into primary parts. We show that the smooth concordance group of topologically slice knots has a large subgroup for which the conjectures are true and there are infinitely many primary parts, each of which has infinite rank. This supports the conjectures for topologically slice knots. We also prove analogues for the associated graded groups of the bipolar filtration of topologically slice knots. Among ingredients of the proof, we use amenable $L^2$ -signatures, Ozsváth-Szabó d-invariants and Némethi’s result on Heegaard Floer homology of Seifert 3-manifolds. In an appendix, we present a general formulation of the notion of primary decomposition.

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A survey on Heegaard Floer homology and concordance

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216517400156 ◽

2017 ◽

Vol 26 (02) ◽

pp. 1740015 ◽

Cited By ~ 10

Author(s):

Jennifer Hom

Keyword(s):

Floer Homology ◽

Heegaard Floer Homology ◽

Survey Paper ◽

Stable Equivalence ◽

Knot Concordance ◽

Heegaard Floer ◽

The Way

In this survey paper, we discuss several different knot concordance invariants coming from the Heegaard Floer homology package of Ozsváth and Szabó. Along the way, we prove that if two knots are concordant, then their knot Floer complexes satisfy a certain type of stable equivalence.

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