# A necessary condition for two string links to have the same closure up to concordance

## Summary (1 min read)

### 1. Introduction

- In this paper the authors study link concordance via string links.
- In their paper [1] the authors provided a connection between the two approaches, obtaining a certain diagram of groups that distinguishes links 620 Mathematics Subject Classification.

### 2. Ambient isotopy

- This product induces a monoid structure on the set SLðkÞ of isotopy classes of k-string links.
- Therefore m0ðbÞn induces a homomorphism G0ðbÞn : F ð2kÞ F ð2kÞn hai~aiF ð2kÞniN !.
- PðbÞ pðbÞn hyi ~yipðbÞniN : G0ðbÞn pðbÞ pðbÞn hyi ~yipðbÞniN !ðG1ðbÞnÞ 1 Fð2kÞ F ð2kÞn hai~aiF ð2kÞniN ðIIÞ where the vertical maps are quotient maps.
- Conversely suppose the authors have a commutative diagram (*) with bn an automorphism.

### 3. Concordance

- Habegger-Lin’s actions induce actions CSLð2kÞ CSLðkÞ !.
- GFðkÞ such that diagram (**) is commutative.

### 4. Correction

- After making these changes the authors will have the converse in Theorem 8 there.
- José Eduardo Prado Pires de Campos Instituto de Ciências Matemáticas e de Computação Universidade de São Paulo Caixa Postal 668, 13560-970 São Carlos, SP Brazil E-mail: jeppc@icmc.usp.br.

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### Additional excerpts

...By Stallings’ Theorem [14] (see also [6]), they also induce isomorphisms on the lower central series quotients of fundamental groups: FðkÞ F ðkÞn ! ðm0ð f ÞÞn G pð f Þ pð f Þn ðm1ð f ÞÞn G F ðkÞ FðkÞn :...

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...By Stallings’ Theorem [14] (see also [6]), they also induce isomorphisms on the lower central series quotients of fundamental groups: FðkÞ F ðkÞn !ðm0ð f ÞÞn G pð f Þ pð f Þn ðm1ð f ÞÞn G F ðkÞ FðkÞn : Therefore ðm0ð f ÞÞnðm1ð f ÞÞ 1 n is an element Anð f Þ A Aut FðkÞ FðkÞn , the group of automorphisms of FðkÞ FðkÞn ....

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### "A necessary condition for two strin..." refers result in this paper

...Its study, as well as the study of link concordance, continued in the works of Cappell and Shaneson [3] Tristram [15], Levine ([9], [10]), Ko [8] and others....

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...[15] A. G. Tristram, Some cobordism invariants for links, Proc....

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