Cobordism of satellite knots

Litherland, R. A.

doi:10.1090/conm/035/780587

Cited by 66 publications

(69 citation statements)

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“…This follows from the additivity of Casson-Gordon invariants; see [Litherland 1984] or [Gilmer 1983]. The only unexpected aspect of the formula is that, since we are dealing with K J # −K r J , it might have been anticipated that the difference (a)s 7 (J ) − (b)s 7 (J ) would appear rather than the sum.…”

Section: The Basic Examplesmentioning

confidence: 99%

Knot mutation: 4-genus of knots and algebraic concordance

Kim¹,

Livingston²

2005

Pacific J. Math.

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Kearton observed that mutation can change the concordance class of a knot. A close examination of his example reveals that it is of 4-genus 1 and has a mutant of 4-genus 0. The first goal of this paper is to show by examples that for any pair of nonnegative integers m and n there is a knot of 4-genus m with a mutant of 4-genus n.A second result is a crossing change formula for the algebraic concordance class of a knot, which is then applied to prove the invariance of the algebraic concordance class under mutation. We conclude with an application of crossing change formulas to give a short new proof of Long's theorem that strongly positive amphicheiral knots are algebraically slice.

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Section: The Basic Examplesmentioning

confidence: 99%

Knot mutation: 4-genus of knots and algebraic concordance

Kim¹,

Livingston²

2005

Pacific J. Math.

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“…In our examples we use a satellite construction to get knots whose eta invariants can be computed explicitly by methods introduced by Litherland [Lit84].…”

Section: Summary Of Resultsmentioning

confidence: 99%

Eta invariants as sliceness obstructions and their relation to Casson–Gordon invariants

Friedl

2004

Algebr. Geom. Topol.

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We give a useful classification of the metabelian unitary representations of π 1 (M K ), where M K is the result of zero-surgery along a knot K ⊂ S 3 . We show that certain eta invariants associated to metabelian representations π 1 (M K ) → U (k) vanish for slice knots and that even more eta invariants vanish for ribbon knots and doubly slice knots. We show that our vanishing results contain the Casson-Gordon sliceness obstruction. In many cases eta invariants can be easily computed for satellite knots. We use this to study the relation between the eta invariant sliceness obstruction, the eta-invariant ribbonness obstruction, and the L 2 -eta invariant sliceness obstruction recently introduced by Cochran, Orr and Teichner. In particular we give an example of a knot which has zero eta invariant and zero metabelian L 2 -eta invariant sliceness obstruction but which is not ribbon.

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“…Notice that at least one of χ i and χ j is nontrivial. Furthermore, since according to [1] (see also [6]) the set of characters for which the Casson-Gordon invariants must vanish is a metabolizer for the linking form on Litherland's analysis [14] of companionship and Casson-Gordon invariants applies directly to the case of knotting the bands in the Seifert surface (see also [7])). Roughly stated, there is a correspondence between characters on H 1 (M (K)) and on H 1 (M (K i )); it then follows that the difference of the corresponding Casson-Gordon invariants is determined by q -signatures of…”

Section: Proof Of Theorem 11mentioning

confidence: 99%