Geodesic
Slice-torus Concordance Invariants and Whitehead Doubles of Links
Canadian journal of mathematics, Tome 72 (2020) no. 6, pp. 1423-1462

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/S0008414X19000294
Mots-clés : Link concordance, slice-torus invariant, splitting number 
Cavallo, Alberto; Collari, Carlo. Slice-torus Concordance Invariants and Whitehead Doubles of Links. Canadian journal of mathematics, Tome 72 (2020) no. 6, pp. 1423-1462. doi: 10.4153/S0008414X19000294
@article{10_4153_S0008414X19000294,
     author = {Cavallo, Alberto and Collari, Carlo},
     title = {Slice-torus {Concordance} {Invariants} and {Whitehead} {Doubles} of {Links}},
     journal = {Canadian journal of mathematics},
     pages = {1423--1462},
     year = {2020},
     volume = {72},
     number = {6},
     doi = {10.4153/S0008414X19000294},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X19000294/}
}
TY  - JOUR
AU  - Cavallo, Alberto
AU  - Collari, Carlo
TI  - Slice-torus Concordance Invariants and Whitehead Doubles of Links
JO  - Canadian journal of mathematics
PY  - 2020
SP  - 1423
EP  - 1462
VL  - 72
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.4153/S0008414X19000294/
DO  - 10.4153/S0008414X19000294
ID  - 10_4153_S0008414X19000294
ER  - 
%0 Journal Article
%A Cavallo, Alberto
%A Collari, Carlo
%T Slice-torus Concordance Invariants and Whitehead Doubles of Links
%J Canadian journal of mathematics
%D 2020
%P 1423-1462
%V 72
%N 6
%U http://geodesic.mathdoc.fr/articles/10.4153/S0008414X19000294/
%R 10.4153/S0008414X19000294
%F 10_4153_S0008414X19000294

[1] Abe, T., The Rasmussen invariant of a homogeneous knot. Proc. Amer. Math. Soc. 139(2011), 2647–2656. https://doi.org/10.1090/S0002-9939-2010-10687-1 Google Scholar | DOI

[2] Adams, C. C., Splitting versus unlinking. J. Knot Theory Ramifications 5(1996), no. 3, 295–299. https://doi.org/10.1142/S0218216596000205 Google Scholar | DOI

[3] Bar-Natan, D., Khovanov’s homology for tangles and cobordisms. Geom. Topol. 9(2005), 1443–1499. https://doi.org/10.2140/gt.2005.9.1443 Google Scholar | DOI

[4] 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-2008-053-1 Google Scholar | DOI

[5] 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/06831047 Google Scholar | DOI

[6] 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/S0218216515500212 Google Scholar

[7] 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.2018.18.1917 Google Scholar | DOI

[8] Cavallo, A., On Bennequin type inequalities for links in tight contact 3-manifolds. Google Scholar

[9] 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/S0013091516000420 Google Scholar | DOI

[10] 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/13156 Google Scholar | DOI

[11] Collari, C., Transverse invariants from the deformations of Khovanov - and -homologies. PhD thesis, Università degli studi di Firenze, 2017. Google Scholar

[12] Collari, C., A Bennequin-type inequality and combinatorial bounds. Michigan Math. J., to appear. Google Scholar

[13] Conway, A., Invariants of colored links and generalizations of the Burau representation. PhD thesis, Université de Genève, 2017. Google Scholar

[14] Etnyre, J., Legendrian and transversal knots. Handbook of knot theory. Elsevier B. V., Amsterdam, 2005, pp. 105–185. https://doi.org/10.1016/B978-044451452-3/50004-6 Google Scholar | DOI

[15] Hedden, M., Knot Floer homology of Whitehead doubles. Geom. Topol. 11(2007), 2277–2338. https://doi.org/10.2140/gt.2007.11.2277 Google Scholar | DOI

[16] 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.2015.05.034 Google Scholar | DOI

[17] 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

[18] Jeong, G., A family of link concordance invariants from perturbed homology. Google Scholar

[19] Lee, E. S., An endomorphism of the Khovanov invariant. Adv. Math. 197(2005), no. 2, 554–586. https://doi.org/10.1016/j.aim.2004.10.015 Google Scholar | DOI

[20] Lewark, L., Rasmussen’s spectral sequence and the concordance invariants. Adv. Math. 260(2014), 59–83. https://doi.org/10.1016/j.aim.2014.04.003 Google Scholar | DOI

[21] Livingston, C., Computations of the Ozsváth–Szabó concordance invariant. Compos. Math. 147(2011), 661–668. Google Scholar

[22] 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.2006.6.651 Google Scholar | DOI

[23] Lobb, A. J., A slice genus lower bound from Khovanov–Rozansky homology. Adv. Math. 222(2009), no. 4, 1220–1276. https://doi.org/10.1016/j.aim.2009.06.001 Google Scholar | DOI

[24] Lobb, A. J., Computable bounds for Rasmussen’s concordance invariant. Compos. Math. 147(2011), no. 2, 661–668. https://doi.org/10.1112/S0010437X10005117 Google Scholar | DOI

[25] 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.2012.12.293 Google Scholar | DOI

[26] 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/S0218216507005312 Google Scholar | DOI

[27] 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.2003.05.001 Google Scholar | DOI

[28] Park, J., Inequality on t (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/13306 Google Scholar | DOI

[29] Rasmussen, J., Khovanov homology and the slice genus. Invent. Math. 182(2010), no. 2, 419–447. https://doi.org/10.1007/s00222-010-0275-6 Google Scholar | DOI

[30] Rolfsen, D., Knots and links. Publish or Perish Inc., Houston, 1990, pp. xiv + 439. Google Scholar

[31] 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-C Google Scholar | DOI

[32] 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/S0218216592000227 Google Scholar | DOI

[33] 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/S0273-0979-1993-00397-5 Google Scholar | DOI

[34] Wolfram Research, Inc., Mathematica, Version 11.3. Champaign, IL, 2018. Google Scholar

[35] 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.2008.12.003 Google Scholar | DOI

Cité par Sources :