Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-25T10:35:59.724Z Has data issue: false hasContentIssue false
  • English
  • Français

The ϒ function of L–space knots is a Legendre transform

Published online by Cambridge University Press:  20 March 2017

MACIEJ BORODZIK  and
MATTHEW HEDDEN
MACIEJ BORODZIK
Affiliation:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland. e-mail: mcboro@mimuw.edu.pl
MATTHEW HEDDEN
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, U.S.A. e-mail: mhedden@math.msu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Given an L–space knot we show that its ϒ function is the Legendre transform of a counting function equivalent to the d–invariants of its large surgeries. The unknotting obstruction obtained for the ϒ function is, in the case of L–space knots, contained in the d–invariants of large surgeries. Generalisations apply for connected sums of L–space knots, which imply that the slice obstruction provided by ϒ on the subgroup of concordance generated by L–space knots is no finer than that provided by the d–invariants.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 3 , May 2018 , pp. 401 - 411
Copyright
Copyright © Cambridge Philosophical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Baader, S. Unknotting sequences for torus knots. Math. Proc. Camb. Phil. Soc. 148 (2010), no. 1, 111–116.Google Scholar
[2] Baader, S. Scissor equivalence for torus links. Bull. Lond. Math. Soc. 44 (2012), 1068–1078.Google Scholar
[3] Baader, S. Bipartite graphs and quasipositive surfaces. Quart. J. Math. 65 (2014), 655–664.Google Scholar
[4] Baker, K. A note on the concordance of fibered knots. J. Topol. 9 (2016), no. 1, 1–4.Google Scholar
[5] Borodzik, M. and Livingston, C. Heegaard Floer homologies and rational cuspidal curves. Forum of Math. Sigma, 2 (2014), e28.Google Scholar
[6] Borodzik, M. and Livingston, C. Semigroups, d-invariants and deformations of cuspidal singular points of plane curves. J. Lond. Math. Soc. 93 (2016), no. 2, 439–463.Google Scholar
[7] Eisenbud, D. and Neumann, W. Three-dimensional link theory and invariants of plane curve singularities. Annals of Mathematics Studies, vol. 110 (Princeton University Press, 1985).Google Scholar
[8] Feller, P. Gordian adjacency for torus knots. Algebr. Geom. Topol. 14 (2014), 769–793.CrossRefGoogle Scholar
[9] Feller, P. Minimal cobordisms between torus knots. Preprint 2015, arxiv:1501.00483.Google Scholar
[10] Feller, P. and Krcatovich, D. On cobordisms between knots, braid index and the Upsilon-invariant. Preprint 2016, arxiv:1602.02637.Google Scholar
[11] Frøyshov, K. An inequality for the h-invariant in instanton Floer theory. Topology 43 2004, no.2, 407–432.Google Scholar
[12] Hedden, M. On knot Floer homology and cabling. Ph.D. thesis. Columbia University (2005).Google Scholar
[13] Hedden, M. Notions of positivity and the Ozsváth-Szabó concordance invariant. J. Knot Theory Ramifications 19 (2010), no. 5, 617–629.Google Scholar
[14] Hedden, M., Kirk, P. and Livingston, C. Non-slice linear combinations of algebraic knots. J. Eur. Math. Soc. 14 (2012), no. 4, 1181–1208.Google Scholar
[15] Hedden, M., Livingston, C. and Ruberman, D. Topologically slice knots with nontrivial Alexander polynomial. Adv. Math. 231 (2012), 913–939.Google Scholar
[16] Hom, J. On the concordance genus of topologically slice knots. Int. Math. Res. Not. (2015) no. 5, 1295–1314.Google Scholar
[17] Krcatovich, D. The reduced Floer complex, Topology Appl. 194 (2015), 171–201.Google Scholar
[18] Kronheimer, P. and Mrowka, T. Gauge theory for embedded surfaces, I. Topology 32 (1993), no. 4, 773–826.Google Scholar
[19] Livingston, C. Notes about the υ function. Preprint 2014, arxiv: 1412.0254.Google Scholar
[20] Micallef, M. and White, B. The structure of branch points in minimal surfaces and in pseudoholomorphic curves. Ann. of Math. (2) 141 (1995), no. 1, 35–85.Google Scholar
[21] Milnor, J. Singular points of complex hypersurfaces. Ann. Math. Stu. vol. 61 (Princeton University Press, 1968).Google Scholar
[22] Ni, Y. and Wu, Z. Cosmetic surgeries on knots in S 3. J. Reine Angew. Math. 706 (2015), 1–17.Google Scholar
[23] Owens, B. and Strle, S. Immersed disks, slicing numbers and concordance unknotting numbers. Preprint 2013, arxiv:1311.6702.Google Scholar
[24] Ozsváth, P. and Szabó, Z. Holomorphic disks and knot invariants. Adv. Math. 186 (2004), no. 1, 58–116.Google Scholar
[25] Ozsváth, P., Szabó, Z. and Stipsicz, A. Concordance homomorphisms from knot Floer homology. Preprint 2014, arxiv: 1407.1795.Google Scholar
[26] Ozsváth, P. and Szabó, Z. Knot Floer homology and lens space surgeries. Topology 44 (2005), 1281–1300.Google Scholar
[27] Ozsváth, P. and Szabó, Z. Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary. Adv. Math. 186 (2004), 58–116.Google Scholar
[28] Peters, T. A concordance invariant from the Floer homology of ±1 surgeries. Preprint, arxiv: 1003.3038.Google Scholar
[29] Petkova, I. Cables of thin knots and bordered Heegaard Floer homology. Quantum Topol. 4 (4) (2013), 377–409.Google Scholar
[30] Rasmussen, J. Floer homology and knot complements. Ph.D. thesis. Harvard University (2003).Google Scholar
[31] Rasmussen, J. Lens space surgeries and a conjecture of Goda and Teragaito. Geom. Topol. 8 (2004), 1013–1031.Google Scholar
[32] Rockafellar, R. Convex analysis. Princeton Mathematical Series, No. 28 (Princeton University Press, Princeton, NJ, 1970).Google Scholar
[33] Rudolph, L. Quasipositivity as an obstruction to sliceness. Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 51–59.Google Scholar
[34] Wall, C. Singular points of plane curves. London Math. Soc. Student Texts, 63 (Cambridge University Press, Cambridge, 2004).Google Scholar
[35] Wang, S. The genus filtration in the smooth concordance group. Pacific J. Math. 285 (2016), no. 2, 501–510.Google Scholar