Research ArticleNo Access

Shake slice and shake concordant links

Andrews University, 4260 Administration Drive, Berrien Springs, Michigan USA

https://doi.org/10.1142/S021821652050087XCited by:1 (Source: Crossref)

Abstract

We can construct a $4$ $4$ -manifold by attaching $2$ $2$ -handles to a $4$ $4$ -ball with framing $r$ $r$ along the components of a link in the boundary of the $4$ $4$ -ball. We define a link as $r$ $r$ -shake slice if there exists embedded spheres that represent the generators of the second homology of the $4$ $4$ -manifold. This naturally extends $r$ $r$ -shake slice, a generalization of slice that has previously only been studied for knots, to links of more than one component. We also define a relative notion of shake $r$ $r$ -concordance for links and versions with stricter conditions on the embedded spheres that we call strongly $r$ $r$ -shake slice and strongly $r$ $r$ -shake concordance. We provide infinite families of links that distinguish concordance, shake concordance, and strong shake concordance. Moreover, for $r = 0$ $r = 0$ we completely characterize shake slice and shake concordant links in terms of concordance and string link infection. This characterization allows us to prove that the first non-vanishing Milnor $\bar{μ}$ $\bar{μ}$ invariants are invariants of shake concordance. We also argue that shake concordance does not imply link homotopy.

Keywords:

AMSC: 57M25

References

1. T. Abe, I. D. Jong, Y. Omae and M. Takeuchi, Annulus twist and diffeomorphic 4-manifolds, Math. Proc. Cambridge Philos. Soc. 155(2) (2013) 219–235. Crossref, Web of Science, Google Scholar
2. S. Akbulut, On 2-dimensional homology classes of 4-manifolds, Math. Proc. Cambridge Philos. Soc. 82(1) (1977) 99–106. Crossref, Web of Science, Google Scholar
3. S. Akbulut, Knots and exotic smooth structures on 4-manifolds, J. Knot Theory Ramifications 2(1) (1993) 1–10. Link, Google Scholar
4. A. J. Casson, Link cobordism and milnor’s invariant, Bull. London Math. Soc. 7(1) (1975) 39–40. Crossref, Google Scholar
5. J. C. Cha, T. Kim, D. Ruberman and S. Strle, Smooth concordance of links topologically concordant to the Hopf link, Bull. Lond. Math. Soc. 44(3) (2012) 443–450. Crossref, Web of Science, Google Scholar
6. T. D. Cochran, Derivatives of Links: Milnor’s Concordance Invariants and Massey’s Products, Vol. 427 (American Mathematical Society, 1990). Google Scholar
7. T. Cochran, S. Friedl and P. Teichner, New constructions of slice links, Comment. Math. Helv. 84(3) (2009) 617–638. Crossref, Web of Science, Google Scholar
8. T. D. Cochran and A. Ray, Shake slice and shake concordant knots, J. Topol. 9(3) (2016) 861–888. Crossref, Web of Science, Google Scholar
9. C. Giffen, Link concordance implies link homotopy, Math. Scand. 45(2) (1979) 243–254. Crossref, Web of Science, Google Scholar
10. D. Goldsmith, Concordance implies homotopy for classical links in $M^{3}$ $M^{3}$ , Comment. Math. Helv. 54(1) (1979) 347–355. Crossref, Web of Science, Google Scholar
11. H. Jin Jang, M. Hoon Kim, M. Powell et al., Smooth slice boundary links whose derivative links have nonvanishing milnor invariants, Michigan Math. J. 63(2) (2014) 423–446. Web of Science, Google Scholar
12. J. P. Levine, An approach to homotopy classification of links, Trans. American Math. Soc. 306(1) (1988) 361–387. Crossref, Web of Science, Google Scholar
13. J. Milnor, Link groups, Ann. Math. 59(2) (1954) 177–195. Crossref, Web of Science, Google Scholar
14. J. Milnor, Isotopy of links, Algebr. Geom. Topol. (1957) 280–306. Google Scholar
15. P. Ozsváth and Z. Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003) 615–639. Crossref, Web of Science, Google Scholar
16. W. B. Raymond Lickorish, Shake–slice knots, Topology of Low-Dimensional Manifolds (Springer, 1979), pp. 67–70. Crossref, Google Scholar