Winding number $m$ and $-m$ patterns acting on concordance
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- by Allison N. Miller PDF
- Proc. Amer. Math. Soc. 147 (2019), 2723-2731 Request permission
Abstract:
We prove that for any winding number $m>0$ pattern $P$ and winding number $-m$ pattern $Q$, there exist knots $K$ such that the minimal genus of a cobordism between $P(K)$ and $Q(K)$ is arbitrarily large. This answers a question posed by Cochran-Harvey [Algebr. Geom. Topol. 18 (2018), pp. 2509–2540] and generalizes a result of Kim-Livingston [Pacific J. Math. 220 (2005), pp. 87–105].References
- 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
- Tim Cochran and Shelly Harvey, The geometry of the knot concordance space, Algebr. Geom. Topol. 18 (2018), no. 5, 2509–2540. MR 3848393, DOI 10.2140/agt.2018.18.2509
- Jae Choon Cha and Charles Livingston, Knot signature functions are independent, Proc. Amer. Math. Soc. 132 (2004), no. 9, 2809–2816. MR 2054808, DOI 10.1090/S0002-9939-04-07378-2
- Patrick M. Gilmer, On the slice genus of knots, Invent. Math. 66 (1982), no. 2, 191–197. MR 656619, DOI 10.1007/BF01389390
- Paul Kirk and Charles Livingston, Twisted knot polynomials: inversion, mutation and concordance, Topology 38 (1999), no. 3, 663–671. MR 1670424, DOI 10.1016/S0040-9383(98)00040-8
- Se-Goo Kim and Charles Livingston, Knot mutation: 4-genus of knots and algebraic concordance, Pacific J. Math. 220 (2005), no. 1, 87–105. MR 2195064, DOI 10.2140/pjm.2005.220.87
- J. Levine, A characterization of knot polynomials, Topology 4 (1965), 135–141. MR 180964, DOI 10.1016/0040-9383(65)90061-3
- R. A. Litherland, Cobordism of satellite knots, Four-manifold theory (Durham, N.H., 1982) Contemp. Math., vol. 35, Amer. Math. Soc., Providence, RI, 1984, pp. 327–362. MR 780587, DOI 10.1090/conm/035/780587
Additional Information
- Allison N. Miller
- Affiliation: Department of Mathematics, Rice University, 6100 Main Street, Houston, Texas 77005
- MR Author ID: 999009
- Received by editor(s): January 17, 2018
- Received by editor(s) in revised form: September 23, 2018
- Published electronically: March 7, 2019
- Communicated by: David Futer
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 2723-2731
- MSC (2010): Primary 57M25, 57M27
- DOI: https://doi.org/10.1090/proc/14439
- MathSciNet review: 3951445