A lower bound for the double slice genus
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Abstract:
In this paper, we develop a lower bound for the double slice genus of a knot using Casson-Gordon invariants. As an application, we show that the double slice genus can be arbitrarily larger than twice the slice genus. As an analogue to the double slice genus, we also define the superslice genus of a knot, and give both an upper bound and a lower bound in the topological category.References
- Paolo Aceto, Marco Golla, and Kyle Larson, Embedding 3-manifolds in spin 4-manifolds, J. Topol. 10 (2017), no. 2, 301–323. MR 3653313, DOI 10.1112/topo.12010
- A. J. Casson and C. McA. Gordon, Cobordism of classical knots, À la recherche de la topologie perdue, Progr. Math., vol. 62, Birkhäuser Boston, Boston, MA, 1986, pp. 181–199. With an appendix by P. M. Gilmer. MR 900252
- Peter Feller, The degree of the Alexander polynomial is an upper bound for the topological slice genus, Geom. Topol. 20 (2016), no. 3, 1763–1771. MR 3523068, DOI 10.2140/gt.2016.20.1763
- Peter Feller and Lukas Lewark, On classical upper bounds for slice genera, Selecta Math. (N.S.) 24 (2018), no. 5, 4885–4916. MR 3874707, DOI 10.1007/s00029-018-0435-x
- P. Feller and L. Lewark. Balanced algebraic unknotting, linking forms, and surfaces in three-and four-space. arXiv preprint arXiv:1905.08305, 2019.
- R. H. Fox, Some problems in knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 168–176. MR 0140100
- Michael H. Freedman, The disk theorem for four-dimensional manifolds, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 647–663. MR 804721
- Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR 1201584
- Stefan Friedl, Eta invariants as sliceness obstructions and their relation to Casson-Gordon invariants, Algebr. Geom. Topol. 4 (2004), 893–934. MR 2100685, DOI 10.2140/agt.2004.4.893
- Patrick M. Gilmer, Configurations of surfaces in $4$-manifolds, Trans. Amer. Math. Soc. 264 (1981), no. 2, 353–380. MR 603768, DOI 10.1090/S0002-9947-1981-0603768-7
- Patrick M. Gilmer and Charles Livingston, On embedding $3$-manifolds in $4$-space, Topology 22 (1983), no. 3, 241–252. MR 710099, DOI 10.1016/0040-9383(83)90011-3
- C. McA. Gordon and D. W. Sumners, Knotted ball pairs whose product with an interval is unknotted, Math. Ann. 217 (1975), no. 1, 47–52. MR 380816, DOI 10.1007/BF01363239
- Bo Ju Jiang, A simple proof that the concordance group of algebraically slice knots is infinitely generated, Proc. Amer. Math. Soc. 83 (1981), no. 1, 189–192. MR 620010, DOI 10.1090/S0002-9939-1981-0620010-7
- Louis H. Kauffman and Laurence R. Taylor, Signature of links, Trans. Amer. Math. Soc. 216 (1976), 351–365. MR 388373, DOI 10.1090/S0002-9947-1976-0388373-0
- Taehee Kim, New obstructions to doubly slicing knots, Topology 45 (2006), no. 3, 543–566. MR 2218756, DOI 10.1016/j.top.2005.11.005
- Charles Livingston and Jeffrey Meier, Doubly slice knots with low crossing number, New York J. Math. 21 (2015), 1007–1026. MR 3425633
- Jeffrey Meier, Distinguishing topologically and smoothly doubly slice knots, J. Topol. 8 (2015), no. 2, 315–351. MR 3356764, DOI 10.1112/jtopol/jtu027
- Patrick Orson, Double $L$-groups and doubly slice knots, Algebr. Geom. Topol. 17 (2017), no. 1, 273–329. MR 3604378, DOI 10.2140/agt.2017.17.273
- Daniel Ruberman, Doubly slice knots and the Casson-Gordon invariants, Trans. Amer. Math. Soc. 279 (1983), no. 2, 569–588. MR 709569, DOI 10.1090/S0002-9947-1983-0709569-5
- Daniel Ruberman, On smoothly superslice knots, New York J. Math. 22 (2016), 711–714. MR 3548119
- D. W. Sumners, Invertible knot cobordisms, Comment. Math. Helv. 46 (1971), 240–256. MR 290351, DOI 10.1007/BF02566842
Additional Information
- Wenzhao Chen
- Affiliation: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
- MR Author ID: 1309566
- Email: chenwenz@msu.edu
- Received by editor(s): January 24, 2020
- Received by editor(s) in revised form: April 30, 2020, and May 2, 2020
- Published electronically: February 2, 2021
- Additional Notes: The author is grateful to the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support.
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 2541-2558
- MSC (2020): Primary 57K10, 57K31
- DOI: https://doi.org/10.1090/tran/8191
- MathSciNet review: 4223025