×

A note on L-spaces which are double branched covers of non-quasi-alternating links. (English) Zbl 1376.57002

Quasi-alternating knots and links were introduced by P. Ozsváth and Z. Szabó [Adv. Math. 194, No. 1, 1–33 (2005; Zbl 1076.57013)] in their Heegaard Floer homology theory, and they are now recognized as one of the important classes of knots. Such links are homologically thin, but the converse statement is known to be false by J. Greene [Math. Res. Lett. 17, No. 1, 39–49 (2010; Zbl 1232.57008)]. The simplest example is the non-alternating Montesinos knot \(11_{n50}\) in the knot table, which is a Kanenobu knot. In fact, Greene gives an infinite family of homologically thin, non-quasi-alternating links \(L_{m,n}\). These links are Montesinos links with three tangles. Let \(X_{m,n}\) be the double branched cover of the \(3\)-sphere over the link \(L_{m,n}\), which is a small Seifert fibered space.
The purpose of the paper under review is to show that this double cover \(X_{m,n}\) is an \(L\)-space for any \(n>m\geq 2\) and that \(X_{m,n}\) cannot be obtained as a double branched cover of another link. The first part follows from the criterion for Seifert fibered spaces to be \(L\)-spaces in terms of Seifert invariants. The proof of the second part occupies most of the paper.

MSC:

57M12 Low-dimensional topology of special (e.g., branched) coverings
57M25 Knots and links in the \(3\)-sphere (MSC2010)

Software:

HFK; Khoho
PDFBibTeX XMLCite
Full Text: DOI

References:

[2] Birman, J. S.; Gonzáles-Acuña, F.; Montesinos, J. M., Heegaard splittings of prime 3-manifolds are not unique, Mich. Math. J., 23, 97-103 (1976) · Zbl 0321.57004
[3] Bonahon, F.; Siebenmann, L., New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, freely available at · Zbl 0571.57011
[4] Boyer, S.; Rolfsen, D.; Wiest, B., Orderable 3-manifold groups, Ann. Inst. Fourier, 55, 243-288 (2005) · Zbl 1068.57001
[5] Dunfield, N.; Hoffman, N.; Licata, J., Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling, Math. Res. Lett., 22, 1679-1698 (2015) · Zbl 1351.57022
[6] Eisenbud, D.; Hirsch, U.; Neumann, W., Transverse foliations on Seifert bundles and self-homeomorphisms of the circle, Comment. Math. Helv., 56, 638-660 (1981) · Zbl 0516.57015
[7] Gordon, C. McA.; Lidman, T., Taut foliations, left-orderability, and cyclic branched covers, Acta Math. Vietnam., 39, 599-635 (2014) · Zbl 1310.57023
[8] Greene, J. E., Homologically thin, non-quasi-alternating links, Math. Res. Lett., 17, 1, 39-49 (2009) · Zbl 1232.57008
[9] Greene, J. E., Lattices, graphs, and Conway mutation, Invent. Math., 192, 717-750 (2012) · Zbl 1278.57021
[10] Greene, J. E., Conway mutation and alternating links, (Proc. 18th Gökova Geometry-Topology Conference (2011)), 31-41 · Zbl 1360.57016
[11] Greene, J. E.; Watson, L., Turaev torsion, definite 4-manifolds, and quasi-alternating knots, Bull. Lond. Math. Soc., 45, 962-972 (2013) · Zbl 1420.57021
[12] Hatcher, A. E., Notes on basic 3-manifold topology, freely available at
[13] Hodgson, C.; Rubinstein, J. H., Involutions and isotopies of lens spaces, (Knot Theory and Manifolds. Knot Theory and Manifolds, Vancouver, B.C., 1983. Knot Theory and Manifolds. Knot Theory and Manifolds, Vancouver, B.C., 1983, Lecture Notes in Math., vol. 1144 (1985), Springer: Springer Berlin), 60-96
[14] Jankins, M.; Neumann, W., Rotation number and products of circle homomorphisms, Math. Ann., 271, 381-400 (1985) · Zbl 0543.57019
[15] Jiang, B.; Wang, S.; Wu, Y.-Q., Homeomorphisms of 3-manifolds and the realization of Nielsen number, Commun. Anal. Geom., 9, 825-877 (2001) · Zbl 1015.55002
[16] Kawauchi, A., Classification of pretzel knots, Kobe J. Math., 2, 11-22 (1985) · Zbl 0586.57005
[17] Kim, P., Involutions on prism manifolds, Trans. Am. Math. Soc., 268, 377-409 (1982)
[18] Lisca, P.; Stipsicz, A., Ozsváth-Szabó invariants and tight contact 3-manifolds, III, J. Symplectic Geom., 5, 357-384 (2007) · Zbl 1149.57037
[19] Meeks, W.; Scott, P., Finite group actions on 3-manifolds, Invent. Math., 86, 287-346 (1986) · Zbl 0626.57006
[20] Miyazaki, K.; Motegi, K., Seifert fibering surgery on periodic knots, Topol. Appl., 121, 275-285 (2002) · Zbl 1004.57004
[21] Montesinos, J. M., Variedades de Seifert que son recubridores cíclicos ramificados de dos hojas, Bol. Soc. Mat. Mexicana, 18, 1-32 (1973)
[22] Motegi, K., Dehn surgeries, group actions and Seifert fiber spaces, Commun. Anal. Geom., 11, 343-389 (2003) · Zbl 1086.57017
[23] Motegi, K., L-space surgery and twisting operation, Algebraic Geom. Topol., 16, 1727-1772 (2016) · Zbl 1345.57015
[24] Naimi, R., Foliations transverse to fibers of Seifert manifolds, Comment. Math. Helv., 69, 155-162 (1994) · Zbl 0797.55009
[25] Neumann, W. D.; Raymond, F., Seifert manifolds, plumbing, \(μ\)-invariant and orientation reversing maps, (Algebraic and Geometric Topology, Proc. Sympos., Univ. California. Algebraic and Geometric Topology, Proc. Sympos., Univ. California, Santa Barbara, Calif., 1977. Algebraic and Geometric Topology, Proc. Sympos., Univ. California. Algebraic and Geometric Topology, Proc. Sympos., Univ. California, Santa Barbara, Calif., 1977, Lect. Notes in Math., vol. 664 (1978), Springer-Verlag), 161-196
[26] Núñez, V.; Ramírez-Losada, E., The trefoil knot is as universal as it can be, Topol. Appl., 130, 1-17 (2003) · Zbl 1030.57003
[27] Orlik, P., Seifert Manifolds, Lect. Notes in Math., vol. 291 (1972), Springer-Verlag · Zbl 0263.57001
[28] Ozsváth, P.; Szabó, Z., Holomorphic disks and topological invariants for closed three-manifolds, Ann. Math., 159, 1027-1158 (2004) · Zbl 1073.57009
[29] Ozsváth, P.; Szabó, Z., Holomorphic disks and three-manifold invariants: properties and applications, Ann. Math., 159, 1159-1245 (2004) · Zbl 1081.57013
[30] Ozsváth, P.; Szabó, Z., On knot Floer homology and lens space surgeries, Topology, 44, 1281-1300 (2005) · Zbl 1077.57012
[31] Ozsváth, P.; Szabó, Z., On the Heegaard Floer homology of branched double-covers, Adv. Math., 194, 1-33 (2005) · Zbl 1076.57013
[32] Shumakovitch, A., KhoHo pari package (2009)
[33] Song, H. J.; Im, Y. H.; Ko, K. H., LCG moves in crystallizations, Geom. Dedic., 76, 229-251 (1999) · Zbl 0932.57020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.