Classical knot and link concordance

Gilmer, Patrick M.

doi:10.1007/bf02565807

Cited by 32 publications

(57 citation statements)

References 19 publications

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“…These spaces have long played an important role in knot theory, dating back to Reidemeister's use of the linking matrix of a knot's dihedral covering link to distinguish knots with the same Alexander polynomial [21] (see also eg Perko [19]). More recently they have also been used in investigations of knot concordance (eg Gilmer [12]). In addition, branched dihedral covers are useful in 3-manifold topology: for example, it turns out that every 3-manifold is a 3-fold branched dihedral covering space over some knot (see eg Burde and Zieschang [2,Theorem 11.11]).…”

Section: Introductionmentioning

confidence: 99%

Surgery presentations of coloured knots and of their covering links

Kricker

Moskovich

2009

Algebr. Geom. Topol.

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We consider knots equipped with a representation of their knot groups onto a dihedral group D 2n (where n is odd). To each such knot there corresponds a closed 3-manifold, the (irregular) dihedral branched covering space, with the branching set over the knot forming a link in it. We report a variety of results relating to the problem of passing from the initial data of a D 2n -coloured knot to a surgery presentation of the corresponding branched covering space and covering link. In particular, we describe effective algorithms for constructing such presentations. A by-product of these investigations is a proof of the conjecture that two D 2n -coloured knots are related by a sequence of surgeries along˙1-framed unknots in the kernel of the representation if and only if they have the same coloured untying invariant (a Z n -valued algebraic invariant of D 2n -coloured knots). 57M12; 57M25

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Section: Introductionmentioning

confidence: 99%

Surgery presentations of coloured knots and of their covering links

Kricker

Moskovich

2009

Algebr. Geom. Topol.

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“…In [8], the group of characters on the first homology of the d-fold cyclic cover of S 3 branched along K is identified with a certain subgroup A d of H 1 (F ) ⊗ Q/Z. In [14], it is proved that for an equivariant slice knot K, there is an equivariant metabolizer H in the above sense such that if d is a prime power order and χ ∈ A d ∩ H ⊗ Q/Z is a character of prime power order, then the Casson-Gordon invariant τ (K, χ) vanish.…”

Section: Introductionmentioning

confidence: 99%

On equivariant slice knots

Choon

1999

Proc. Amer. Math. Soc.

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“…In the case of slice knots, it first appeared in the unpublished Ph.D. thesis of D. Cooper and can also be compared to [Gi,Theorem 4]. It is not known if the latter theorem implies ours or vice-versa.…”

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confidence: 97%

“…Moreover this implies that K is algebraically slice. The following algebraic fact about a genus 1 algebraically slice knot can be found in [Gi,Section 5]. A is a cyclic module with precisely 2 proper submodules each generated by a simple closed curve on F of zero self-linking number.…”

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confidence: 99%