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Transactions of the American Mathematical Society

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On the topological 4-genus of torus knots


Authors: S. Baader, P. Feller, L. Lewark and L. Liechti
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 57M25
DOI: https://doi.org/10.1090/tran/7051
Published electronically: December 19, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus knots with non-maximal signature invariant.


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  • [Baa12] Sebastian Baader, Scissor equivalence for torus links, Bull. Lond. Math. Soc. 44 (2012), no. 5, 1068-1078. MR 2975163, https://doi.org/10.1112/blms/bds044
  • [BL15] Sebastian Baader and Lukas Lewark: The stable 4-genus of alternating knots, 2015, to appear in Asian J. Math. arXiv:1505.03345.
  • [Col15] J. Collins, Seifert matrix computation, retrieved 2015. A computer program to compute Seifert matrices from braids, available from http://www.maths.ed.ac.uk/~jcollins/SeifertMatrix/.
  • [FM12] Benson Farb and Dan Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR 2850125
  • [Fel16] 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, https://doi.org/10.2140/gt.2016.20.1763
  • [FM15] Peter Feller and Duncan McCoy, On 2-bridge knots with differing smooth and topological slice genera, Proc. Amer. Math. Soc. 144 (2016), no. 12, 5435-5442. MR 3556284
  • [FQ90] Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR 1201584
  • [Fre82] Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no. 3, 357-453. MR 679066
  • [GT04] Stavros Garoufalidis and Peter Teichner, On knots with trivial Alexander polynomial, J. Differential Geom. 67 (2004), no. 1, 167-193. MR 2153483
  • [GLM81] C. McA. Gordon, R. A. Litherland, and K. Murasugi, Signatures of covering links, Canad. J. Math. 33 (1981), no. 2, 381-394. MR 617628, https://doi.org/10.4153/CJM-1981-032-3
  • [KT76] Louis H. Kauffman and Laurence R. Taylor, Signature of links, Trans. Amer. Math. Soc. 216 (1976), 351-365. MR 0388373, https://doi.org/10.2307/1997704
  • [KM94] P. B. Kronheimer and T. S. Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994), no. 6, 797-808. MR 1306022, https://doi.org/10.4310/MRL.1994.v1.n6.a14
  • [LW97] Ronnie Lee and Dariusz M. Wilczyński, Representing homology classes by locally flat surfaces of minimum genus, Amer. J. Math. 119 (1997), no. 5, 1119-1137. MR 1473071
  • [Lie16] Livio Liechti, Signature, positive Hopf plumbing and the Coxeter transformation, Osaka J. Math. 53 (2016), no. 1, 251-266. With an appendix by Peter Feller and Liechti. MR 3466832
  • [Liv10] Charles Livingston, The stable 4-genus of knots, Algebr. Geom. Topol. 10 (2010), no. 4, 2191-2202. MR 2745668, https://doi.org/10.2140/agt.2010.10.2191
  • [Mur65] Kunio Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387-422. MR 0171275, https://doi.org/10.2307/1994215
  • [PAR15] The PARI Group, Bordeaux: Pari/gp, version 2.7.4, 2015. Programming library available from http://pari.math.u-bordeaux.fr/.
  • [Pow16] Mark Powell, The four-genus of a link, Levine-Tristram signatures and satellites, J. Knot Theory Ramifications 26 (2017), no. 2, 1740008, 28. MR 3604490, https://doi.org/10.1142/S0218216517400089
  • [Rud82] Lee Rudolph, Nontrivial positive braids have positive signature, Topology 21 (1982), no. 3, 325-327. MR 649763, https://doi.org/10.1016/0040-9383(82)90014-3
  • [Rud84] Lee Rudolph, Some topologically locally-flat surfaces in the complex projective plane, Comment. Math. Helv. 59 (1984), no. 4, 592-599. MR 780078, https://doi.org/10.1007/BF02566368
  • [Rud93] Lee Rudolph, Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 51-59. MR 1193540, https://doi.org/10.1090/S0273-0979-1993-00397-5
  • [Sta78] John R. Stallings, Constructions of fibred knots and links, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 55-60. MR 520522
  • [Tri69] A. G. Tristram, Some cobordism invariants for links, Proc. Cambridge Philos. Soc. 66 (1969), 251-264. MR 0248854

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Additional Information

S. Baader
Affiliation: Mathematisches Institut, Universität Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
Email: sebastian.baader@math.unibe.ch

P. Feller
Affiliation: Department of Mathematics, Boston College, Maloney Hall, Chestnut Hill, Massachusetts 02467
Address at time of publication: ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
Email: peter.feller@math.ch

L. Lewark
Affiliation: Mathematisches Institut, Universität Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
Email: lukas.lewark@math.unibe.ch

L. Liechti
Affiliation: Mathematisches Institut, Universität Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
Email: livio.liechti@math.unibe.ch

DOI: https://doi.org/10.1090/tran/7051
Received by editor(s): October 19, 2015
Received by editor(s) in revised form: June 20, 2016, and August 2, 2016
Published electronically: December 19, 2017
Additional Notes: The third author thanks the EPSRC grant EP/K00591X/1 for providing computing facilities
The second, third and fourth authors gratefully acknowledge support by the SNSF grants 155477 and 159208, respectively
Article copyright: © Copyright 2017 by the authors

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