Links not concordant to the Hopf link

Abstract

We give new Casson-Gordon style obstructions for a two-component link to be topologically concordant to the Hopf link.1 Lemma 3.2. Let p, q be primes. For any two-component link J withProof. Since the linking number is 1, the inclusion maps ∂X 1 J → X J are Z-homology equivalences. We wish to apply Proposition 3.1 with S = S 1 × S 1 to see that H * (X ϕ J ; Q(ξ q k )(H)) ∼ = H * ((∂X 1 J ) ϕ ; Q(ξ q k )(H)) ∼ = 0.

Concordance of links with identical Alexander invariants

“…To find such representations, first we consider links that are 'big' in the sense that they admit representations into non-abelian nilpotent quotients. It is straightforward to show (see The case of small links resembles known approaches to the study of knot concordance [2,9] and it is related to earlier works of the authors [3,20].…”

Nonconcordant links with homology cobordant zero-framed surgery manifolds

“…There are many linking number one 2-component links which are not concordant, as can be detected, for example, by the multivariable Alexander polynomial [Kawauchi 1978;Nakagawa 1978]. For related in-depth study, the reader is referred to, for instance, [Cha and Ko 1999;Friedl and Powell 2011;Cha 2014]. With our respective coauthors, we gave nonconcordant linking number one links with two unknotted components, for which abelian invariants such as the multivariable Alexander polynomial are unable to obstruct them from being concordant.…”

“…Our results serve to underline the philosophy that when investigating the relative problem of whether two links are concordant, and neither of them are the unlink, one should consider obstructions to homology cobordism of the link exteriors viewed as bordered manifolds, rather than to homology cobordism of the zero-surgery manifolds, even in low dimensions. This was implemented in, for example, [Kawauchi 1978;Nakagawa 1978;Cha 2014] (see also [Friedl and Powell 2011] for a related approach).…”

An Injectivity Theorem for Casson-Gordon Type Representations Relating to the Concordance of Knots and Links

“…In a future application [11] we will consider the case where Y is the exterior of a 2-component link with linking number one and S is one of the two boundary tori.…”

“…This allows the possibility of using representations of link complements of Casson-Gordon type, i.e., representations which do not necessarily factor through nilpotent quotients of the link group. We use this fact to give in [11] a new Casson-Gordon type obstruction to a link being concordant to a Hopf link: for a preprint see [11].…”