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Casson towers and slice links

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Abstract

We prove that a Casson tower of height 4 contains a flat embedded disc bounded by the attaching circle, and we prove disc embedding results for height 2 and 3 Casson towers which are embedded into a 4-manifold, with some additional fundamental group assumptions. In the proofs we create a capped grope from a Casson tower and use a refined height raising argument to establish the existence of a symmetric grope which has two layers of caps, data which is sufficient for a topological disc to exist, with the desired boundary. As applications, we present new slice knots and links by giving direct applications of the disc embedding theorem to produce slice discs, without first constructing a complementary 4-manifold. In particular we construct a family of slice knots which are potential counterexamples to the homotopy ribbon slice conjecture.

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Notes

  1. In fact, Freedman showed that a two component link called “Whitehead\({}_3\)” bounds slicing discs in the 4-ball whose complement has free fundamental group. This link is associated to the simplest Casson tower T of height 3, as explained in our Sect. 6.1. It turns out that each of the two slicing discs is ambiently isotopic to the standard disc in the 4-ball by [20, 11.7 A]. It follows that the exterior of one slicing disc is T and the other slicing disc is bounded by C(T).

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Acknowledgments

The authors would like to thank Kent Orr and Peter Teichner for some very helpful conversations and suggestions. Wojciech Politarczyk and Mark Powell worked together on the combinatorics chapter for the Freedman lecture notes, from which the proof of Lemma 3.7 is derived, and Mark Powell gained a great deal of understanding from this collaboration. We also thank the referees for their very useful comments. Jae Choon Cha was partially supported by NRF grants 2013067043 and 2013053914.

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Authors and Affiliations

  1. Department of Mathematics, POSTECH, Pohang, 790-784, Republic of Korea

    Jae Choon Cha

  2. School of Mathematics, Korea Institute for Advanced Study, Seoul, 130-722, Republic of Korea

    Jae Choon Cha

  3. Départment de Mathématiques, UQAM, Montreal, QC, Canada

    Mark Powell

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  1. Jae Choon Cha
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Cha, J.C., Powell, M. Casson towers and slice links. Invent. math. 205, 413–457 (2016). https://doi.org/10.1007/s00222-015-0639-z

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