Linear Independence of Knots Arising from Iterated Infection Without the Use of Tristram–Levine Signature

Davis, Chelsea S.

doi:10.1093/imrn/rns277

Cited by 3 publications

(1 citation statement)

References 18 publications

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“…Before proving the theorem, we note that, by happenstance the specific examples used in [14,Theorem 3.3] do lie in B n , and so that theorem may be directly applied without modification to show the slightly weaker result that for n ≥ 1 there is an infinite rank subgroup…”

Section: Nontriviality Of the B Filtrationmentioning

confidence: 99%

Filtering smooth concordance classes of topologically slice knots

2013

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We propose and analyze a structure with which to organize the difference between a knot in S 3 bounding a topologically embedded 2-disk in B 4 and it bounding a smoothly embedded disk. The n-solvable filtration of the topological knot concordance group, due to Cochran-Orr-Teichner, may be complete in the sense that any knot in the intersection of its terms may well be topologically slice. However, the natural extension of this filtration to what is called the n-solvable filtration of the smooth knot concordance group, is unsatisfactory because any topologically slice knot lies in every term of the filtration. To ameliorate this we investigate a new filtration, fB n g, that is simultaneously a refinement of the n-solvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. We show that each B n =B nC1 has infinite rank. But our primary interest is in the induced filtration, fT n g, on the subgroup, T , of knots that are topologically slice. We prove that T =T 0 is large, detected by gauge-theoretic invariants and the , s , -invariants, while the nontriviality of T 0 =T 1 can be detected by certain d -invariants. All of these concordance obstructions vanish for knots in T 1 . Nonetheless, going beyond this, our main result is that T 1 =T 2 has positive rank. Moreover under a "weak homotopy-ribbon" condition, we show that each T n =T nC1 has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity. 57M25

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Section: Nontriviality Of the B Filtrationmentioning

confidence: 99%

Filtering smooth concordance classes of topologically slice knots

2013

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Twisted Blanchfield Pairings and Symmetric Chain Complexes

Powell

2016

Q J Math

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We define the twisted Blanchfield pairing of a symmetric triad of chain complexes over a group ring Z[π], together with a unitary representation of π over an Ore domain with involution. We prove that the pairing is sesquilinear, and we prove that it is hermitian and nonsingular under certain extra conditions. A twisted Blanchfield pairing is then associated to a 3-manifold together with a decomposition of its boundary into two pieces and a unitary representation of its fundamental group.T H 1 (N ; Z) × T H 1 (N ; Z) → Q/Z.

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Amenable signatures, algebraic solutions and filtrations of the knot concordance group

Kim

2020

Algebr. Geom. Topol.

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Linear Independence of Knots Arising from Iterated Infection Without the Use of Tristram–Levine Signature

Cited by 3 publications

References 18 publications

Filtering smooth concordance classes of topologically slice knots

Filtering smooth concordance classes of topologically slice knots

Twisted Blanchfield Pairings and Symmetric Chain Complexes

Amenable signatures, algebraic solutions and filtrations of the knot concordance group

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