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The four-genus of connected sums of torus knots

Published online by Cambridge University Press:  17 April 2017

CHARLES LIVINGSTON  and
CORNELIA A. VAN COTT
CHARLES LIVINGSTON
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405, U.S.A. e-mail: livingst@indiana.edu
CORNELIA A. VAN COTT
Affiliation:
Department of Mathematics and Statistics, University of San Francisco, San Francisco, CA 94117, U.S.A e-mail: cvancott@usfca.edu
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Abstract

We study the four-genus of linear combinations of torus knots: g4(aT(p, q) #-bT(p′, q′)). Fixing positive p, q, p′, and q′, our focus is on the behavior of the four-genus as a function of positive a and b. Three types of examples are presented: in the first, for all a and b the four-genus is completely determined by the Tristram–Levine signature function; for the second, the recently defined Upsilon function of Ozsváth–Stipsicz–Szabó determines the four-genus for all a and b; for the third, a surprising interplay between signatures and Upsilon appears.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 3 , May 2018 , pp. 531 - 550
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[1] Baader, S. Scissor equivalence for torus links. Bull. Lond. Math. Soc. 44 (2012), 1068–1078.CrossRefGoogle Scholar
[2] Baader, S. Unknotting sequences for torus knots. Math. Proc. Camb. Philos. Soc. 148 (2010), 111–116.Google Scholar
[3] Baader, S., Feller, P., Lewark, L. and Liechti, L. On the topological 4-genus of torus knots. Trans. Amer. Math. Soc. (To appear).Google Scholar
[4] Borodzik, M. and Hedden, M. The Upsilon function of L-space knots is a Legendre transform. arXiv:1505.06672.Google Scholar
[5] Borodzik, M. and Livingston, C. Semigroups, d-invariants and deformations of cuspidal singular points of plane curves. J. Lond. Math. Soc. (2) 93 (2016), 439–463.Google Scholar
[6] Feller, P. Gordian adjacency for torus knots. Alg. Geom. Topol. 14 (2014), 769–793.Google Scholar
[7] Feller, P. Optimal cobordisms between torus knots. Comm. Anal. Geom. (To appear).Google Scholar
[8] Feller, P., Pohlmann, S. and Zentner, R. Alternating numbers of torus knots with small braid index. Indiana Univ. Math. J. (To appear).Google Scholar
[9] Gilmer, P. and Livingston, C. Signature jumps and Alexander polynomials of links. Proc. Amer. Math. Soc. 144 (2016), 5407–5417.Google Scholar
[10] Goldsmith, D. Symmetric fibered links in Knots, Groups and 3–Manifolds. Ann. of Math. Studies 84, ed. L. P. Neuwirth, (Princeton 1975).Google Scholar
[11] Hom, J. and Wu, Z. Four-ball genus bounds and a refinement of the Ozsváth-Szabó tau-invariant. J. Symplectic Geom. 14 (2016), 305–323.Google Scholar
[12] Kronheimer, P. and Mrowka, T. Gauge theory for embedded surfaces. I. Topology 32 (1993), 773–826.Google Scholar
[13] Levine, J. Invariants of knot cobordism. Invent. Math. 8 (1969), 98–110.Google Scholar
[14] Livingston, C. The stable 4-genus of knots. Alg. Geom. Topol. 10 (2010), 2191–2202.Google Scholar
[15] Livingston, C. Notes on the knot concordance invariant Upsilon. Alg. Geom. Topol. 17 (2017), 111–130.Google Scholar
[16] Litherland, R. Signatures of iterated torus knots. In Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), Lecture Notes in Math. 722, (Springer, Berlin, 1979), 71–84.Google Scholar
[17] Milnor, J. Infinite cyclic covers. In Topology of Manifolds, Complementary Series in Mathematics vol. 13, ed. Hocking, J. G. (Prindle, Weber & Schmidt. Boston, 1968).Google Scholar
[18] Murasugi, K. Knot theory and its applications, Birkhauser Boston, 1996.Google Scholar
[19] Owens, B. and Strle, S. Immersed disks, slicing numbers and concordance unknotting numbers. arXiv:1311.6702.Google Scholar
[20] Ozsváth, P. and Szabó, Z. Knot Floer homology and the four-ball genus. Geom. Topol. 7 (2003), 615–639.Google Scholar
[21] Ozsváth, P., Stipsicz, A. and Szabó, Z. Concordance homomorphisms from knot Floer homology. arXiv:1407.1795.Google Scholar
[22] Ozsváth, P., Stipsicz, A. and Szabó, Z. Unoriented knot Floer homology and the unoriented four-ball genus. Int. Math. Res. Notices (2016), rnw143.Google Scholar
[23] Rasmussen, J. Khovanov homology and the slice genus. Invent. Math. 182 (2010), 419–447.Google Scholar
[24] Rudolph, L. Some topologically locally-flat surfaces in the complex projective plane. Comment. Math. Helv. 59 (1984), 592–599.Google Scholar
[25] Rudolph, L. Quasipositivity as an obstruction to sliceness. Bull. Amer. Math. Soc. 29 (1993), 51–59.Google Scholar
[26] Tristram, A. Some cobordism invariants for links. Proc. Camb. Phil. Soc. 66 (1969), 251–264.Google Scholar