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Characterization of Positive Links and the s-invariant for Links

Published online by Cambridge University Press:  20 November 2018

Tetsuya Abe  and
Keiji Tagami
Tetsuya Abe
Affiliation:
Osaka City University Advanced Mathematical Institute, 3-3-138 Sugimoto, Sumiyoshi-ku Osaka 558-8585, Japan e-mail: tabe@sci.osaka-cu.ac.jp
Keiji Tagami
Affiliation:
Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, 378-8510, Japan e-mail: tagami_keiji@ma.noda.tus.ac.jp
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Abstract

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We characterize positive links in terms of strong quasipositivity, homogeneity, and the value of Rasmussen and Beliakova-Wehrli's $s$-invariant. We also study almost positive links, and in particular, determine the $s$-invariants of almost positive links. This result suggests that all almost positive links might be strongly quasipositive. On the other hand, it implies that almost positive links are never homogeneous links.

Keywords

57M2557M27(almost) positive linkhomogeneous link(strongly) quasipositive links-invariant
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 6 , 01 December 2017 , pp. 1201 - 1218
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Abe, T., The Rasmussen invariant of a homogeneous knot. Proc. Amer. Math. Soc. 139(2011), no. 7, 2647–2656.http://dx.doi.Org/10.1090/S0002-9939-2010-10687-1 Google Scholar
[2] Abe, T., State cycles which represent the canonical class of Lee's homology of a knot. Topology Appl. 159(2012), no. 4, 1146–1158. http://dx.doi.Org/10.1016/j.topol.2011.11.042 Google Scholar
[3] Baader, S., Quasipositivity and homogeneity. Math. Proc. Cambridge Philos. Soc. 139(2005), no. 2, 287–290.http://dx.doi.Org/10.1017/S0305004105008698 Google Scholar
[4] Banks, J. E., Homogeneous links, Seifert surfaces, digraphs and the reduced Alexander polynomial. Geom. Dedicata 166(2013), 67–98. http://dx.doi.org/10.1007/s10711-012-9786-1 Google Scholar
[5] Bar-Natan, D., The knot atlas. http://www.math.toronto.edu/drorbn/KAtlas/ Google Scholar
[6] 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.http://dx.doi.org/10.4153/CJM-2008-053-1 Google Scholar
[7] Cha, J. C. and Livingston, C., Knotlnfo. http://www.indiana.edu/%7eknotinfo/ Google Scholar
[8] Cochran, T. D. and Gompf, R. E., Applications of Donaldson's theorems to classical knot concordance, homology 3-spheres and property P. Topology 27(1988), no. 4, 495–512.http://dx.doi.Org/10.1016/0040-9383(88)90028-6 Google Scholar
[9] Cromwell, P. R., Homogeneous links. J. London Math. Soc. (2) 39(1989), no. 3, 535–552.http://dx.doi.Org/10.1112/jlms/s2-39.3.535 Google Scholar
[10] Gabai, D., The Murasugi sum is a natural geometric operation. In: Low-dimensional topology (San Francisco, Calif., 1981), Contemp. Math., 20, American Mathematical Society, Providence, RI, 1983, pp. 131–143. http://dx.doi.Org/10.1090/conm/020/71 8138 Google Scholar
[11] Gabai, D., The Murasugi sum is a natural geometric operation. II. In: Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982), Contemp. Math., 44, American Mathematical Society, Providence, RI, 1985, pp. 93–100. http://dx.doi.org/10.1090/conm/044/81 3105 Google Scholar
[12] Goda, H., Hirasawa, M., and Yamamoto, R., Almost alternating diagrams andfibered links in S3. Proc. London Math. Soc. (3) 83(2001), no. 2, 472–492.http://dx.doi.Org/10.1112/plms/83.2.472 Google Scholar
[13] Hedden, M., Notions of positivity and the Ozsváth-Szabó concordance invariant. J. Knot Theory Ramifications 19(2010), no. 5, 617–629. http://dx.doi.org/10.1142/S0218216510008017 Google Scholar
[14] Jong, I. D. and Kishimoto, K., On positive knots of genus two. Kobe J. Math. 30(2013), no. 1-2,1–18.Google Scholar
[15] Kauffman, L. H., Formal knot theory. Mathematical Notes, 30, Princeton University Press, Princeton, NJ, 1983.Google Scholar
[16] Kawamura, T., The Rasmussen invariants and the sharper slice-Bennequin inequality on knots. Topology 46(2007), no. 1, 29–38.http://dx.doi.Org/10.1016/j.top.2006.10.001 Google Scholar
[17] Kawamura, T., An estimate of the Rasmussen invariant for links and the determination for certain links. Topology Appl. 196(2015), 558–574.http://dx.doi.Org/10.1016/j.topol.2015.05.034 Google Scholar
[18] Khovanov, M., A categorification of the Jones polynomial. Duke Math. J. 101(2000), no. 3, 359–426.http://dx.doi.Org/10.1215/S0012-7094-00-10131-7 Google Scholar
[19] Lewark, L., The Rasmussen invariant of arborescent and of mutant links. http://lewark.de/lukas/Master- Lukas- Lewark.pdf Google Scholar
[20] Lickorish, W. B. R., An introduction to knot theory. Graduate Texts in Mathematics, 175, Springer-Verlag, New York, 1997.http://dx.doi.org/10.1007/978-1-4612-0691-0 Google Scholar
[21] Livingston, C., Computations of the Ozsváth-Szabó knot concordance invariant. Geom. Topol. 8(2004), 735–742. http://dx.doi.Org/10.2140/gt.2004.8.735 Google Scholar
[22] Lobb, A., Computable bounds for Rasmussen's concordance invariant. Compos. Math. 147(2011), no. 2, 661–668.http://dx.doi.org/10.1112/S0010437X10005117 Google Scholar
[23] Manchon, P. M. G., Homogeneous links and the Seifert matrix. Pacific J. Math. 255(2012), no. 2, 373–392.http://dx.doi.org/10.2140/pjm.2012.255.373 Google Scholar
[24] Mayland, E. J., Jr. and Murasugi, K., On a structural property of the groups of alternating links. Canad. J. Math. 28(1976), no. 3, 568–588. http://dx.doi.Org/10.4153/CJM-1976-056-8 Google Scholar
[25] Nakamura, T., Four-genus and unknotting number of positive knots and links. Osaka J. Math. 37(2000), no. 2, 441–451.Google Scholar
[26] Nakamura, T., Positive alternating links are positively alternating. J. Knot Theory Ramifications 9(2000), no. 1, 107–112.http://dx.doi.org/10.1142/S0218216500000050 Google Scholar
[27] Ozsváth, P. and Szabo, Z., Knot Floer homology and the four-ball genus. Geom. Topol. 7(2003), 615–639. http://dx.doi.Org/10.2140/gt.2003.7.615 Google Scholar
[28] Ozsváth, P., On the Heegaard Floer homology of branched double-covers. Adv. Math. 194(2005), no. 1, 1–33.http://dx.doi.Org/10.1016/j.aim.2004.05.008 Google Scholar
[29] Plamenevskaya, O., Bounds for the Thurston-Bennequin number from Floer homology. Algebr.Geom. Topol. 4(2004), 399–406. http://dx.doi.org/10.2140/agt.2004.4399 Google Scholar
[30] Plamenevskaya, O., Transverse knots and Khovanov homology. Math. Res. Lett. 13(2006), no. 4, 571–586.http://dx.doi.org/10.4310/MRL.2006.v13.n4.a7 Google Scholar
[31] Przytycki, J. H., Positive knots have negative signature. Bull. Polish Acad. Sci. Math. 37(1989), no. 7-12, 559–562.Google Scholar
[32] Przytycki, J. H.and Taniyama, K., Almost positive links have negative signature. J. Knot Theory Ramifications 19(2010), no. 2, 187–289. http://dx.doi.org/10.1142/S0218216510007838 Google Scholar
[33] Rasmussen, J., Floer homology and knot complements. Ph.D. Thesis, Harvard University, ProQuest LLC, Ann Arbor, MI, 2003.Google Scholar
[34] Rasmussen, J., Khovanov homology and the slice genus. Invent. Math. 182(2010), no. 2, 419–447.http://dx.doi.org/10.1007/s00222-010-0275-6 Google Scholar
[35] Rudolph, L., Constructions of quasipositive knots and links. I. In: Knots, braids and singularities(Plans-sur-Bex, 1982), Monogr. Enseign. Math., 31, Enseignement Math., Geneva, 1983, pp. 233–245.Google Scholar
[36] Rudolph, L., Quasipositive plumbing (constructions of quasipositive knots and links. V). Proc. Amer. Math. Soc. 126(1998), no. 1, 257–267.http://dx.doi.org/10.1090/S0002-9939-98-04407-4 Google Scholar
[37] Rudolph, L., Positive links are strongly quasipositive. In: Proceedings of the Kirbyfest (Berkeley, C A, 1998), Geom. Topol. Monogr., 2, Geom. Topol. Publ., Coventry, 1999, pp. 555–562.http://dx.doi.Org/10.2140/gtm.1999.2.555 Google Scholar
[38] Shumakovitch, A. N., Rasmussen invariant, slice-Bennequin inequality, and sliceness of knots. J. Knot Theory Ramifications 16(2007), no. 10,1403–1412. http://dx.doi.org/10.1142/S0218216507005889 Google Scholar
[39] Silvero, M., On a conjecture by Kauffman on alternative and pseudoalternating links. Topology Appl. 188(2015), 82–90.http://dx.doi.Org/10.1016/j.topol.2015.03.012 Google Scholar
[40] Stoimenow, A., Diagram genus, generators and applications. arxiv:1101.3390 Google Scholar
[41] Stoimenow, A., GaufS diagram sums on almost positive knots. Compos. Math. 140(2004), no. 1, 228–254.http://dx.doi.Org/10.1112/S001 0437X03000174 Google Scholar
[42] Stoimenow, A., On polynomials and surfaces of variously positive links. J. Eur. Math. Soc. (JEMS) 7(2005), no. 4, 477–509.http://dx.doi.Org/10.4171/JEMS/36 Google Scholar
[43] Tagami, K., The Rasmussen invariant, four-genus and three-genus of an almost positive knot are equal. Canad. Math. Bull. 57(2014), no. 2, 431–438.http://dx.doi.Org/10.4153/CMB-2O14-005-7 Google Scholar
[44] Traczyk, P., Nontrivial negative links have positive signature, Manuscripta Math. 61(1988), no. 3, 279–284.http://dx.doi.org/10.1007/BF01258439 Google Scholar
[45] Van Buskirk, J. M., Positive knots have positive Conway polynomials. In: Knot theory and manifolds (Vancouver, B.C., 1983), Lecture Notes in Math., 1144, Springer, Berlin, 1985, pp. 146–159.Google Scholar