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On classical upper bounds for slice genera

Selecta Mathematica volume 24, pages 4885–4916 (2018)Cite this article

Abstract

We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the minimal genus of a surface in the four-ball whose complement has infinite cyclic fundamental group. We characterize the algebraic genus in terms of cobordisms in three-space, and explore the connections to other knot invariants related to the Seifert form, the Blanchfield form, knot genera and unknotting. Employing Casson-Gordon invariants, we discuss the algebraic genus as a candidate for the optimal upper bound for the topological slice genus that is determined by the S-equivalence class of Seifert matrices.

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Acknowledgments

We thank Danny Ruberman for pointing us to [37]. We thank Sebastian Baader and Livio Liechti for valuable inputs; in particular, concerning Proposition 25. We thank Mark Powell for comments on a first version of this paper, and the referee for helpful suggestions. Both authors gratefully acknowledge support by the SNSF and thank the MPIM Bonn for its hospitality.

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Authors and Affiliations

  1. Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092, Zurich, Switzerland

    Peter Feller

  2. Mathematical Institute, University of Bern, Alpeneggstrasse 22, 3012, Bern, Switzerland

    Lukas Lewark

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  1. Peter Feller
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  2. Lukas Lewark
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Correspondence to Lukas Lewark.

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Feller, P., Lewark, L. On classical upper bounds for slice genera. Sel. Math. New Ser. 24, 4885–4916 (2018). https://doi.org/10.1007/s00029-018-0435-x

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  • DOI: https://doi.org/10.1007/s00029-018-0435-x

Keywords

  • Slice genus
  • Seifert form
  • Casson-Gordon invariants
  • Algebraic unknotting number

Mathematics Subject Classification

  • 57M25
  • 57M27