A lower bound for the double slice genus
Author:
Wenzhao Chen
Journal:
Trans. Amer. Math. Soc. 374 (2021), 2541-2558
MSC (2020):
Primary 57K10, 57K31
DOI:
https://doi.org/10.1090/tran/8191
Published electronically:
February 2, 2021
MathSciNet review:
4223025
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Abstract | References | Similar Articles | Additional Information
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.
- Paolo Aceto, Marco Golla, and Kyle Larson, Embedding 3-manifolds in spin 4-manifolds, J. Topol. 10 (2017), no. 2, 301β323. MR 3653313, DOI https://doi.org/10.1112/topo.12010
- A. J. Casson and C. McA. Gordon, Cobordism of classical knots, Γ la recherche de la topologie perdue, Progr. Math., vol. 62, BirkhΓ€user Boston, Boston, MA, 1986, pp. 181β199. 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, 1763β1771. MR 3523068, DOI https://doi.org/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, 4885β4916. MR 3874707, DOI https://doi.org/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. 168β176. 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. 647β663. 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), 893β934. MR 2100685, DOI https://doi.org/10.2140/agt.2004.4.893
- Patrick M. Gilmer, Configurations of surfaces in $4$-manifolds, Trans. Amer. Math. Soc. 264 (1981), no. 2, 353β380. MR 603768, DOI https://doi.org/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, 241β252. MR 710099, DOI https://doi.org/10.1016/0040-9383%2883%2990011-3
- C. McA. Gordon and D. W. Sumners, Knotted ball pairs whose product with an interval is unknotted, Math. Ann. 217 (1975), no. 1, 47β52. MR 380816, DOI https://doi.org/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, 189β192. MR 620010, DOI https://doi.org/10.1090/S0002-9939-1981-0620010-7
- Louis H. Kauffman and Laurence R. Taylor, Signature of links, Trans. Amer. Math. Soc. 216 (1976), 351β365. MR 388373, DOI https://doi.org/10.1090/S0002-9947-1976-0388373-0
- Taehee Kim, New obstructions to doubly slicing knots, Topology 45 (2006), no. 3, 543β566. MR 2218756, DOI https://doi.org/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), 1007β1026. MR 3425633
- Jeffrey Meier, Distinguishing topologically and smoothly doubly slice knots, J. Topol. 8 (2015), no. 2, 315β351. MR 3356764, DOI https://doi.org/10.1112/jtopol/jtu027
- Patrick Orson, Double $L$-groups and doubly slice knots, Algebr. Geom. Topol. 17 (2017), no. 1, 273β329. MR 3604378, DOI https://doi.org/10.2140/agt.2017.17.273
- Daniel Ruberman, Doubly slice knots and the Casson-Gordon invariants, Trans. Amer. Math. Soc. 279 (1983), no. 2, 569β588. MR 709569, DOI https://doi.org/10.1090/S0002-9947-1983-0709569-5
- Daniel Ruberman, On smoothly superslice knots, New York J. Math. 22 (2016), 711β714. MR 3548119
- D. W. Sumners, Invertible knot cobordisms, Comment. Math. Helv. 46 (1971), 240β256. MR 290351, DOI https://doi.org/10.1007/BF02566842
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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.
Article copyright:
© Copyright 2021
American Mathematical Society