scholarly journals On independence of iterated Whitehead doubles in the knot concordance group

2018 ◽  
Vol 27 (01) ◽  
pp. 1850003
Author(s):  
Kyungbae Park
Keyword(s):  
Correction Term ◽  
Khovanov Homology ◽  
Floer Theory ◽  
Knot Concordance ◽  
Torus Knot ◽  
Linearly Independent ◽  
Heegaard Floer

Let [Formula: see text] be the positively clasped untwisted Whitehead double of a knot [Formula: see text], and [Formula: see text] be the [Formula: see text] torus knot. We show that [Formula: see text] and [Formula: see text] are linearly independent in the smooth knot concordance group [Formula: see text] for each [Formula: see text]. Further, [Formula: see text] and [Formula: see text] generate a [Formula: see text] summand in the subgroup of [Formula: see text] generated by topologically slice knots. We use the concordance invariant [Formula: see text] of Manolescu and Owens, using Heegaard Floer correction term. Interestingly, these results are not easily shown using other concordance invariants such as the [Formula: see text]-invariant of knot Floer theory and the [Formula: see text]-invariant of Khovanov homology. We also determine the infinity version of the knot Floer complex of [Formula: see text] for any [Formula: see text] generalizing a result for [Formula: see text] of Hedden, Kim and Livingston.

scholarly journals LINEAR INDEPENDENCE IN THE RATIONAL HOMOLOGY COBORDISM GROUP

2019 ◽  
pp. 1-12 ◽  
Author(s):  
Marco Golla ◽  
Kyle Larson
Keyword(s):  
Infinite Order ◽  
Double Cover ◽  
Rational Homology ◽  
Cobordism Group ◽  
Connected Sums ◽  
Knot Concordance ◽  
Linearly Independent ◽  
Rational Homology Spheres ◽  
Homology Cobordism ◽  
Correction Terms

We give simple homological conditions for a rational homology 3-sphere $Y$ to have infinite order in the rational homology cobordism group $\unicode[STIX]{x1D6E9}_{\mathbb{Q}}^{3}$ , and for a collection of rational homology spheres to be linearly independent. These translate immediately to statements about knot concordance when $Y$ is the branched double cover of a knot, recovering some results of Livingston and Naik. The statements depend only on the homology groups of the 3-manifolds, but are proven through an analysis of correction terms and their behavior under connected sums.

scholarly journals Knots having the same Seifert form and primary decomposition of knot concordance

2017 ◽  
Vol 26 (14) ◽  
pp. 1750103
Author(s):  
Taehee Kim
Keyword(s):  
Infinite Family ◽  
Alexander Polynomial ◽  
Primary Decomposition ◽  
Amenable Groups ◽  
Von Neumann ◽  
Knot Concordance ◽  
Linearly Independent

We show that for each Seifert form of an algebraically slice knot with nontrivial Alexander polynomial, there exists an infinite family of knots having the Seifert form such that the knots are linearly independent in the knot concordance group and not concordant to any knot with coprime Alexander polynomial. Key ingredients for the proof are Cheeger–Gromov–von Neumann [Formula: see text]-invariants for amenable groups developed by Cha–Orr and polynomial splittings of metabelian [Formula: see text]-invariants.

scholarly journals The decategorification of bordered Khovanov homology

2014 ◽  
Vol 23 (14) ◽  
pp. 1450078 ◽  
Author(s):  
Lawrence P. Roberts
Keyword(s):  
Floer Homology ◽  
Jones Polynomial ◽  
Type A ◽  
Khovanov Homology ◽  
Heegaard Floer Homology ◽  
Type D ◽  
Heegaard Floer

In [A type D structure in Khovanov homology, preprint (2013), arXiv:1304.0463; A type A structure in Khovanov homology, preprint (2013), arXiv:1304.0465] the author constructed a package which describes how to decompose the Khovanov homology of a link ℒ into the algebraic pairing of a type D structure and a type A structure (as defined in bordered Floer homology), whenever a diagram for ℒ is decomposed into the union of two tangles. Since Khovanov homology is the categorification of a version of the Jones polynomial, it is natural to ask what the types A and D structures categorify, and how their pairing is encoded in the decategorifications. In this paper, the author constructs the decategorifications of these two structures, in a manner similar to I. Petkova's decategorification of bordered Floer homology, [The decategorification of bordered Heegaard-Floer homology, preprint (2012), arXiv:1212.4529v1], and shows how they recover the Jones polynomial. We also use the decategorifications to compare this approach to tangle decompositions with M. Khovanov's from [A functor-valued invariant of tangles, Algebr. Geom. Topol.2 (2002) 665–741].

scholarly journals Combinatorial cobordism maps in hat Heegaard Floer theory

2008 ◽  
Vol 145 (2) ◽  
pp. 207-247 ◽  
Author(s):  
Robert Lipshitz ◽  
Ciprian Manolescu ◽  
Jiajun Wang
Keyword(s):  
Floer Theory ◽  
Heegaard Floer

ALEXANDER POLYNOMIALS AND ORDERS OF HOMOLOGY GROUPS OF BRANCHED COVERS OF KNOTS

2009 ◽  
Vol 18 (07) ◽  
pp. 973-984 ◽  
Author(s):  
SE-GOO KIM
Keyword(s):  
Prime Power ◽  
Alexander Polynomial ◽  
Branched Covers ◽  
Splitting Property ◽  
Homology Groups ◽  
Alexander Polynomials ◽  
Knot Concordance ◽  
Branched Cover ◽  
Linearly Independent

Fox showed that the order of homology of a cyclic branched cover of a knot is determined by its Alexander polynomial. We find examples of knots with relatively prime Alexander polynomials such that the first homology groups of their q-fold cyclic branched covers are of the same order for every prime power q. Furthermore, we show that these knots are linearly independent in the knot concordance group using the polynomial splitting property of the Casson–Gordon–Gilmer invariants.

scholarly journals Tau invariants for balanced spatial graphs

2020 ◽  
Vol 29 (09) ◽  
pp. 2050066
Author(s):  
Katherine Vance
Keyword(s):  
Lower Bound ◽  
Floer Homology ◽  
Chain Complex ◽  
Homology Theory ◽  
Heegaard Floer Homology ◽  
Combinatorial Proof ◽  
Knot Concordance ◽  
Spatial Graphs ◽  
Heegaard Floer ◽  
Definition Of

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.

scholarly journals Knot concordance and Heegaard Floer homology invariants in branched covers

2008 ◽  
Vol 12 (4) ◽  
pp. 2249-2275 ◽  
Author(s):  
J Elisenda Grigsby ◽  
Daniel Ruberman ◽  
Sašo Strle
Keyword(s):  
Floer Homology ◽  
Heegaard Floer Homology ◽  
Branched Covers ◽  
Knot Concordance ◽  
Heegaard Floer

scholarly journals Primary decomposition in the smooth concordance group of topologically slice knots

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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