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A ribbon obstruction and derivatives of knots

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Abstract

We define an obstruction for a knot to be ℤ[ℤ]-homology ribbon, and use this to provide restrictions on the integers that can occur as the triple linking numbers of derivative links of knots that are either homotopy ribbon or doubly slice. Our main application finds new non-doubly slice knots. In particular, this gives new information on the doubly solvable filtration of Taehee Kim: doubly algebraically slice ribbon knots need not be doubly (1)-solvable, and doubly algebraically slice knots need not be (0.5, 1)-solvable. We introduce a notion of homotopy (1)-solvable and find a knot that is (0.5)-solvable but not homotopy (1)-solvable. We also discuss potential connections to unsolved conjectures in knot concordance, such as generalised versions of Kauffman’s conjecture. Moreover, it is possible that our obstruction could fail to vanish on a slice knot.

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Acknowledgements

The first author would like to thank his advisors Tim Cochran and Shelly Harvey, and also Christopher Davis for helpful discussions. The authors also thank Taehee Kim for comments on the first version of this paper. The authors are grateful to the Max Planck Institute for Mathematics and the Hausdorff Institute for Mathematics in Bonn. Part of this paper was written while the authors were visitors at these institutes. The authors respectively thank the Université du Québec à Montréal and Rice University for excellent hospitality. The second author was supported by an NSERC Discovery Grant. Lastly, we are grateful to our anonymous referees for detailed and thoughtful suggestions.

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Authors and Affiliations

  1. Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, 34141, South Korea

    JungHwan Park

  2. Department of Mathematical Sciences, Durham University, Stockton Road, Durham, DH1 3LE, UK

    Mark Powell

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  1. JungHwan Park
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  2. Mark Powell
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Correspondence to Mark Powell.

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Park, J., Powell, M. A ribbon obstruction and derivatives of knots. Isr. J. Math. 250, 265–305 (2022). https://doi.org/10.1007/s11856-022-2338-y

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  • DOI: https://doi.org/10.1007/s11856-022-2338-y

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