Concordances from differences of torus knots to 𝐿-space knots

Allen, Samantha

doi:10.1090/proc/14833

Concordances from differences of torus knots to $L$-space knots
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by Samantha Allen

Proc. Amer. Math. Soc. 148 (2020), 1815-1827

DOI: https://doi.org/10.1090/proc/14833

Published electronically: December 30, 2019

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Abstract:

It is known that connected sums of positive torus knots are not concordant to $L$-space knots. Here we consider differences of torus knots. The main result states that the subgroup of the concordance group generated by two positive torus knots contains no nontrivial $L$-space knots other than the torus knots themselves. Generalizations to subgroups generated by more than two torus knots are also considered.

References

Paolo Aceto and Antonio Alfieri, On sums of torus knots concordant to alternating knots, Bull. Lond. Math. Soc. 51 (2019), no. 2, 327–343. MR 3937591, DOI 10.1112/blms.12228
P. Aceto, D. Celoria, and J. Park, Rational cobordisms and integral homology, arxiv.org/abs/1811.01433.
A. Alfieri, Upsilon type concordance invariants, arxiv.org/abs/1709.01594.
A. Alfieri, S. Kang, and A. I. Stipsicz, Connected Floer homology of covering involutions, arxiv.org/abs/1905.11613.
Maciej Borodzik and Charles Livingston, Heegaard Floer homology and rational cuspidal curves, Forum Math. Sigma 2 (2014), Paper No. e28, 23. MR 3347955, DOI 10.1017/fms.2014.28
Peter Feller and David Krcatovich, On cobordisms between knots, braid index, and the upsilon-invariant, Math. Ann. 369 (2017), no. 1-2, 301–329. MR 3694648, DOI 10.1007/s00208-017-1519-1
Stefan Friedl, Charles Livingston, and Raphael Zentner, Knot concordances and alternating knots, Michigan Math. J. 66 (2017), no. 2, 421–432. MR 3657225, DOI 10.1307/mmj/1491465685
Matthew Hedden and Liam Watson, On the geography and botany of knot Floer homology, Selecta Math. (N.S.) 24 (2018), no. 2, 997–1037. MR 3782416, DOI 10.1007/s00029-017-0351-5
Robion C. Kirby and W. B. Raymond Lickorish, Prime knots and concordance, Math. Proc. Cambridge Philos. Soc. 86 (1979), no. 3, 437–441. MR 542689, DOI 10.1017/S0305004100056280
David Krcatovich, The reduced knot Floer complex, Topology Appl. 194 (2015), 171–201. MR 3404612, DOI 10.1016/j.topol.2015.08.008
R. A. Litherland, Signatures of iterated torus knots, Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977) Lecture Notes in Math., vol. 722, Springer, Berlin, 1979, pp. 71–84. MR 547456
Charles Livingston, Notes on the knot concordance invariant upsilon, Algebr. Geom. Topol. 17 (2017), no. 1, 111–130. MR 3604374, DOI 10.2140/agt.2017.17.111
Charles Livingston, Concordances from connected sums of torus knots to $L$-space knots, New York J. Math. 24 (2018), 233–239. MR 3778502
Robert Myers, Homology cobordisms, link concordances, and hyperbolic $3$-manifolds, Trans. Amer. Math. Soc. 278 (1983), no. 1, 271–288. MR 697074, DOI 10.1090/S0002-9947-1983-0697074-4
Peter Ozsváth and Zoltán Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003), 615–639. MR 2026543, DOI 10.2140/gt.2003.7.615
Peter Ozsváth and Zoltán Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), no. 3, 1027–1158. MR 2113019, DOI 10.4007/annals.2004.159.1027
Peter Ozsváth and Zoltán Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005), no. 6, 1281–1300. MR 2168576, DOI 10.1016/j.top.2005.05.001
Peter S. Ozsváth, András I. Stipsicz, and Zoltán Szabó, Concordance homomorphisms from knot Floer homology, Adv. Math. 315 (2017), 366–426. MR 3667589, DOI 10.1016/j.aim.2017.05.017
Dale Rolfsen, Knots and links, Mathematics Lecture Series, vol. 7, Publish or Perish, Inc., Houston, TX, 1990. Corrected reprint of the 1976 original. MR 1277811
C. T. C. Wall, Singular points of plane curves, London Mathematical Society Student Texts, vol. 63, Cambridge University Press, Cambridge, 2004. MR 2107253, DOI 10.1017/CBO9780511617560
R. Yozgyur, Knots not concordant to L-space knots, arxiv.org/abs/1905.02461.
Ian Zemke, Connected sums and involutive knot Floer homology, Proc. Lond. Math. Soc. (3) 119 (2019), no. 1, 214–265. MR 3957835, DOI 10.1112/plms.12227