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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. C. Blanchfield, Intersection theory of manifolds with operators with applications to knot theory, Annals of Mathematics 65 (1957), 340–356.
K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, Vol. 87, Springer, New York, 1994.
J. R. Burke, Infection by string links and new structure in the knot concordance group, Algebraic & Geometric Topology 14 (2014), 1577–1626.
T. D. Cochran and C. W. Davis, Counterexamples to Kauffman’s conjectures on slice knots, Advances in Mathematics 274 (2015), 263–284.
T. D. Cochran and C. W. Davis, Cut open null-bordisms and derivatives of slice knots, http://arxiv.org/abs/1511.07295.
A. J. Casson and C. M. Gordon, A loop theorem for duality spaces and fibred ribbon knots, Inventiones Mathematicae 74 (1983), 119–137.
T. D. Cochran and R. E. Gompf, Applications of Donaldson’s theorems to classical knot concordance, homology 3-spheres and property P, Topology 27 (1988), 495–512.
T. D. Cochran, S. Harvey and P. Horn, Filtering smooth concordance classes of topologically slice knots, Geometry & Topology 17 (2013), 2103–2162.
T. D. Cochran, S. Harvey and C. Leidy, Knot concordance and higher-order Blanchfield duality, Geometry & Topology 13 (2009), 1419–1482.
T. D. Cochran, S. Harvey and C. Leidy, Derivatives of knots and second-order signatures, Algebraic & Geometric Topology 10 (2010), 739–787.
T. D. Cochran, S. Harvey and C. Leidy, 2-torsion in the n-solvable filtration of the knot concordance group, Proceedings of the London Mathematical Society 102 (2011), 257–290.
J. C. Cha and T. Kim, Unknotted gropes, Whitney towers, and doubly slicing knots, Transactions of the American Mathematical Society 371 (2019), 2383–2429.
T. D. Cochran and W. B. R. Lickorish, Unknotting information from 4-manifolds, Transactions of the American Mathematical Society 297 (1986), 125–142.
T. D. Cochran, Concordance invariance of coefficients of Conway’s link polynomial, Inventiones Mathematicae 82 (1985), 527–541.
T. D. Cochran, Noncommutative knot theory, Algebraic & Geometric Topology 4 (2004), 347–398.
D. Cooper, Signatures of surfaces with applications to knot and link cobordism, Ph. D. thesis, University of Warwick, Coventry, 1982.
T. D. Cochran, K. E. Orr and P. Teichner, Knot concordance, Whitney towers and L2-signatures, Annals of Mathematics 157 (2003), 433–519.
T. D. Cochran, K. E. Orr and P. Teichner, Structure in the classical knot concordance group, Commentarii Mathematici Helvetici 79 (2004), 105–123.
C. W. Davis, T. Martin, C. Otto and J. Park, Every genus one algebraically slice knot is 1-solvable, Transactions of the American Mathematical Society 372 (2019), 3063–3082.
P. M. Gilmer, Slice knots in S3, Quarterly Journal of Mathematics. Oxford 34 (1983), 305–322.
P. M. Gilmer, Classical knot and link concordance, Commentarii Mathematici Helvetici 68 (1993), 1–19.
P. M. Gilmer and C. Livingston, Discriminants of Casson-Gordon invariants, Mathematical Proceedings of the Cambridge Philosophical Society 112 (1992), 127–139.
P. M. Gilmer and C. Livingston, On surgery curves for genus-one slice knots, Pacific Journal of Mathematics 265 (2013), 405–425.
J. Hom and Z. Wu, Four-ball genus bounds and a refinement of the Ozsváth-Szabó tau invariant, Journal of Symplectic Geometry 14 (2016), 305–323.
H. J. Jang, M. H. Kim and M. Powell, Smooth slice boundary links whose derivative links have nonvanishing Milnor invariants, Michigan Mathematical Journal 63 (2014), 423–446.
T. Kim, New obstructions to doubly slicing knots, Topology 45 (2006), 543–566.
J. Levine, Invariants of knot cobordism, Inventiones Mathematicae 8 (1969), 98–110; addendum, ibid. 8 (1969), 355.
J. Levine, Knot modules. I, Transactions of the American Mathematical Society 229 (1977), 1–50.
T. E. Martin, Classification of links up to 0-solvability, Topology and its Applications 317 (2022), Article no. 108158.
J. Milnor, Link groups, Annals of Mathematics 59 (1954), 177–195.
J. Milnor, Isotopy of links. Algebraic geometry and topology, in A symposium in Honor of S. Lefschetz, Princeton University Press, Princeton, NJ, 1957, pp. 280–306.
J. Milnor, Singular Points of Complex Hypersurfaces, Annals of Mathematics Studies, Vol. 61, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1968.
Y. Ni and Z. Wu, Heegaard Floer correction terms and rational genus bounds, Advances in Mathematics 267 (2014), 360–380.
P. Orson, Double L-groups and doubly slice knots, Algebraic & Geometric Topology 17 (2017), 273–329.
P. Ozsváth and Z. Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Advances in Mathematics 173 (2003), 179–261.
P. Ozsváth and Z. Szabó, Knot Floer homology and the four-ball genus, Geometry & Topology 7 (2003), 615–639.
P. Ozsváth, A. I. Stipsicz and Z. Szabó, Concordance homomorphisms from knot Floer homology, Advances in Mathematics 315 (2017), 366–426.
C. Otto, The (n)-solvable filtration of link concordance and Milnor’s invariants, Algebraic & Geometric Topology 14 (2014), 2627–2654.
J. Park, Milnor’s triple linking numbers and derivatives of genus three knots, http://arxiv.org/abs/1603.09163.
T. Peters, A concordance invariant from the Floer homology of +/− 1-surgeries, http://arxiv.org/abs/1003.3038.
H.-C. Wang, The homology groups of the fibre bundles over a sphere, Duke Mathematical Journal 16 (1949), 33–38.
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-022-2338-y