A note on knot concordance and involutive
                    knot Floer homology

Hendricks, Kristen; Hom, Jennifer

doi:10.1090/pspum/102/09

Cited by 3 publications

(4 citation statements)

References 14 publications

(17 reference statements)

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“…In the same paper, they prove that the filtered chain homotopy type of the pair

C F K I^{\infty} false(K false) = false(C F K^{\infty} (K), ι_{K} false)

is an invariant of

K

. In fact, in Hom and Hendricks prove that for

C F K I^{\infty}

an analogue of Theorem holds. Theorem If two knots

K_{1}

and

K_{2}

are concordant, then there exists

double-struckZ

‐graded,

(Z \oplus Z)

‐filtered, acyclic chain complexes

A_{1}

and

A_{2}

together with involutions

ι_{A_{1}}

and

ι_{A_{2}}

such that

false(C F K^{\infty} (K_{1}), ι_{K_{1}} false) \oplus false(A_{1}, ι_{A_{1}} false) ≃ false(C F K^{\infty} (K_{2}), ι_{K_{2}} false) \oplus false(A_{2}, ι_{A_{2}} false)

□

…”

Section: Obstructions From the Kim‐livingston Secondary Invariantmentioning

confidence: 92%

On sums of torus knots concordant to alternating knots

Aceto

Alfieri

2018

Bull. London Math. Soc.

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We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief analysis of the problem in its full generality, we focus on sums of two torus knots. We describe some effective obstructions based on Heegaard Floer homology.be a sum of torus knots. Suppose that p i > q i , and that there is no q i coefficient appearing with repetitions in the list of coefficients ). Denote by Υ a,b (t) the upsilon function of the torus knot T a,b with a > b, thenConsequently, if q i and r i denote, respectively, the quotients and the remainders occurring in the Euclidean algorithm for a and b (so that r 0 = a, r −1 = b, and r i−1 = q i r i + r i+1 ), we have that Υ a,b (t) = n i=0 q i · Υ ri+1,ri (t).

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“…In the same paper, they prove that the filtered chain homotopy type of the pair

C F K I^{\infty} false(K false) = false(C F K^{\infty} (K), ι_{K} false)

is an invariant of

K

. In fact, in Hom and Hendricks prove that for

C F K I^{\infty}

an analogue of Theorem holds. Theorem If two knots

K_{1}

and

K_{2}

are concordant, then there exists

double-struckZ

‐graded,

(Z \oplus Z)

‐filtered, acyclic chain complexes

A_{1}

and

A_{2}

together with involutions

ι_{A_{1}}

and

ι_{A_{2}}

such that

false(C F K^{\infty} (K_{1}), ι_{K_{1}} false) \oplus false(A_{1}, ι_{A_{1}} false) ≃ false(C F K^{\infty} (K_{2}), ι_{K_{2}} false) \oplus false(A_{2}, ι_{A_{2}} false)

□

…”

Section: Obstructions From the Kim‐livingston Secondary Invariantmentioning

confidence: 92%

On sums of torus knots concordant to alternating knots

Aceto

Alfieri

2018

Bull. London Math. Soc.

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“…In light of Proposition 3.11 (see also Corollary 5.1), one might ask whether there are concordance obstructions which can be non-vanishing on knots with V 0 (K) = V 0 (−K) = 0. Recent work of Hendricks and Manolescu [HM15] answers this question in the affirmative. They use the conjugation symmetry on the Heegaard Floer complex to define involutive Heegaard Floer homology, corresponding to Z 4 -equivariant Seiberg-Witten Floer homology.…”

Section: Consider the Chain Mapmentioning

confidence: 96%

A survey on Heegaard Floer homology and concordance

Hom

2017

J. Knot Theory Ramifications

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“…Proof. A result of Hendricks and Hom [1] states that if L is a slice knot, then CFK ∞ (L) splits as the direct sum of involutive complexes: CFK ∞ (L) ∼ = T ⊕ A, where T ∼ = F[U, U −1 ] and A is acyclic. (This result generalizes an analogous result of Hom [4] that holds for noninvolutive complexes.…”

Section: Reductionsmentioning

confidence: 99%

“…We begin by reviewing the definition of the mapping cone complex in the context of the chain map I +I; we denote this complex Cone(C, I +I). The underlying graded vector space is CFK ∞ (K) [1]⊕CFK ∞ (K), where CFK ∞ (K) [1] is the same complex as CFK ∞ (K) with gradings shifted up by 1. The boundary map is given by…”

Section: Introductionmentioning

confidence: 99%

An involutive upsilon knot invariant

Hogancamp,

Livingston

2017

Preprint

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A note on knot concordance and involutive knot Floer homology

Abstract: We prove that if two knots are concordant, then their involutive knot Floer complexes satisfy a certain type of stable equivalence.

Cited by 3 publications

References 14 publications

On sums of torus knots concordant to alternating knots

On sums of torus knots concordant to alternating knots

A survey on Heegaard Floer homology and concordance

An involutive upsilon knot invariant

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