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Slice-torus Concordance Invariants and Whitehead Doubles of Links

Part of: Low-dimensional topology

Published online by Cambridge University Press:  22 May 2019

Alberto Cavallo  and
Carlo Collari
Alberto Cavallo
Affiliation:
Alfréd Rényi Institute of Mathematics, 1053Budapest, Hungary Email: [email protected]@gmail.it
Carlo Collari
Affiliation:
Alfréd Rényi Institute of Mathematics, 1053Budapest, Hungary Email: [email protected]@gmail.it
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Abstract

In this paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to the computation of the splitting number. Finally, we use the slice-torus link invariants and the Whitehead doubling to define new strong concordance invariants for links, which are proven to be independent of the corresponding slice-torus link invariant.

Keywords

Link concordanceslice-torus invariantsplitting number

MSC classification

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57M25: Knots and links in $S^3$
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 6 , December 2020 , pp. 1423 - 1462
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Copyright
© Canadian Mathematical Society 2019

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References

Abe, T., The Rasmussen invariant of a homogeneous knot. Proc. Amer. Math. Soc. 139(2011), 2647–2656. https://doi.org/10.1090/S0002-9939-2010-10687-1CrossRefGoogle Scholar
Adams, C. C., Splitting versus unlinking. J. Knot Theory Ramifications 5(1996), no. 3, 295–299. https://doi.org/10.1142/S0218216596000205CrossRefGoogle Scholar
Bar-Natan, D., Khovanov’s homology for tangles and cobordisms. Geom. Topol. 9(2005), 1443–1499. https://doi.org/10.2140/gt.2005.9.1443CrossRefGoogle Scholar
Beliakova, A. and Wehrli, S., Categorification of the colored Jones polynomial and Rasmussen invariant of links. Canad. J. Math. 60(2008), no. 6, 1240–1266. https://doi.org/10.4153/CJM-2008-053-1CrossRefGoogle Scholar
Borodzik, M., Friedl, S., and Powell, M., Blanchfield forms and Gordian distance. J. Math. Soc. Japan 68(2016), no. 3, 1047–1080. https://doi.org/10.2969/jmsj/06831047CrossRefGoogle Scholar
Cavallo, A., On the slice genus and some concordance invariants of links. J. Knot Theory Ramifications 24(2015), no. 4, 1550021. https://doi.org/10.1142/S0218216515500212Google Scholar
Cavallo, A., The concordance invariant tau in link grid homology. Algebr. Geom. Topol. 18(2018), no. 4, 1917–1951. https://doi.org/10.2140/agt.2018.18.1917CrossRefGoogle Scholar
Cavallo, A., On Bennequin type inequalities for links in tight contact 3-manifolds. arxiv:1801.00614Google Scholar
Cha, J. C., Friedl, S., and Powell, M., Splitting numbers of links. Proc. Edinb. Math. Soc. (2) 60(2017), no. 3, 587–614. https://doi.org/10.1017/S0013091516000420CrossRefGoogle Scholar
Cimasoni, D., Conway, A., and Zacharova, K., Splitting numbers and signatures. Proc. Amer. Math. Soc. 144(2016), no. 12, 5443–5455. https://doi.org/10.1090/proc/13156CrossRefGoogle Scholar
Collari, C., Transverse invariants from the deformations of Khovanov $\mathfrak{sl}_{2}$- and $\mathfrak{sl}_{3}$-homologies. PhD thesis, Università degli studi di Firenze, 2017. https://flore.unifi.it/handle/2158/1079076Google Scholar
Collari, C., A Bennequin-type inequality and combinatorial bounds. Michigan Math. J., to appear. arxiv:1707.03424Google Scholar
Conway, A., Invariants of colored links and generalizations of the Burau representation. PhD thesis, Université de Genève, 2017. http://www.unige.ch/math/folks/conway/ThesisAnthonyConway2.pdfGoogle Scholar
Etnyre, J., Legendrian and transversal knots. Handbook of knot theory. Elsevier B. V., Amsterdam, 2005, pp. 105–185. https://doi.org/10.1016/B978-044451452-3/50004-6CrossRefGoogle Scholar
Hedden, M., Knot Floer homology of Whitehead doubles. Geom. Topol. 11(2007), 2277–2338. https://doi.org/10.2140/gt.2007.11.2277CrossRefGoogle Scholar
Kawamura, T., An estimate of the Rasmussen for links and the determination for certain link. Topology Appl. 192(2015), 558–574. https://doi.org/10.1016/j.topol.2015.05.034CrossRefGoogle Scholar
Kawauchi, A., Shibuya, T., and Suzuki, S., Descriptions on surfaces in four-space. I. Normal forms. Math. Sem. Notes Kobe Univ. 10(1982), no. 1, 75–125.Google Scholar
Jeong, G., A family of link concordance invariants from perturbed $\mathfrak{sl}(n)$homology. arxiv:1608.05781Google Scholar
Lee, E. S., An endomorphism of the Khovanov invariant. Adv. Math. 197(2005), no. 2, 554–586. https://doi.org/10.1016/j.aim.2004.10.015CrossRefGoogle Scholar
Lewark, L., Rasmussen’s spectral sequence and the $\mathfrak{sl}_{n}$concordance invariants. Adv. Math. 260(2014), 59–83. https://doi.org/10.1016/j.aim.2014.04.003CrossRefGoogle Scholar
Livingston, C., Computations of the Ozsváth–Szabó concordance invariant. Compos. Math. 147(2011), 661–668.Google Scholar
Livingston, C. and Naik, S., Ozsváth–Szabó and Rasmussen invariants of doubled knots. Algebr. Geom. Topol. 6(2006), 651–657. https://doi.org/10.2140/agt.2006.6.651CrossRefGoogle Scholar
Lobb, A. J., A slice genus lower bound from $\mathfrak{sl}(n)$Khovanov–Rozansky homology. Adv. Math. 222(2009), no. 4, 1220–1276. https://doi.org/10.1016/j.aim.2009.06.001CrossRefGoogle Scholar
Lobb, A. J., Computable bounds for Rasmussen’s concordance invariant. Compos. Math. 147(2011), no. 2, 661–668. https://doi.org/10.1112/S0010437X10005117CrossRefGoogle Scholar
Lobb, A. J., A note on Gornik’s perturbation of Khovanov–Rozansky homology. Algebr. Geom. Topol. 12(2012), no. 1, 293–305. https://doi.org/10.2140/agt.2012.12.293CrossRefGoogle Scholar
Mackaay, M., Turner, P., and Vaz, P., A remark on Rasmussen’s invariant of knots. J. Knot Theory Ramifications 16(2007), no. 3, 333–344. https://doi.org/10.1142/S0218216507005312CrossRefGoogle Scholar
Ozsváth, P. and Szabó, Z., Holomorphic disks and knot invariants. Adv. Math. 186(2004), no. 1, 58–116. https://doi.org/10.1016/j.aim.2003.05.001CrossRefGoogle Scholar
Park, J., Inequality on t 𝜈(K) defined by Livingston and Naik and its applications. Proc. Amer. Math. Soc. 145(2017), no. 2, 889–891. https://doi.org/10.1090/proc/13306CrossRefGoogle Scholar
Rasmussen, J., Khovanov homology and the slice genus. Invent. Math. 182(2010), no. 2, 419–447. https://doi.org/10.1007/s00222-010-0275-6CrossRefGoogle Scholar
Rolfsen, D., Knots and links. Publish or Perish Inc., Houston, 1990, pp. xiv + 439.Google Scholar
Rudolph, L., Constructions of quasipositive knots and links. III. A characterization of quasipositive Seifert surfaces. Topology 31(1992), no. 2, 231–237. https://doi.org/10.1016/0040-9383(92)90017-CCrossRefGoogle Scholar
Rudolph, L., Quasipositive annuli. (Constructions of quasipositive knots and links. IV). J. Knot Theory Ramifications 1(1992), no. 4, 451–466. https://doi.org/10.1142/S0218216592000227CrossRefGoogle Scholar
Rudolph, L., Quasipositivity as an obstruction to sliceness. Bull. Amer. Math. Soc. (New Ser.) 29(1993), no. 1, 51–59. https://doi.org/10.1090/S0273-0979-1993-00397-5CrossRefGoogle Scholar
Wolfram Research, Inc., Mathematica, Version 11.3. Champaign, IL, 2018.Google Scholar
Wu, H., On the quantum filtration of the Khovanov–Rozansky cohomology. Adv. Math. 221(2009), no. 1, 54–139. https://doi.org/10.1016/j.aim.2008.12.003CrossRefGoogle Scholar