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The Rasmussen Invariant, Four-genus, and Three-genus of an Almost Positive Knot Are Equal

Published online by Cambridge University Press:  20 November 2018

Keiji Tagami*
Affiliation:
Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan e-mail: tagami.k.aa@m.titech.ac.jp
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Abstract

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An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, 4-genus, and 3-genus of a positive knot are equal. In this paper, we prove that the Rasmussen invariant, 4-genus, and 3-genus of an almost positive knot are equal. Moreover, we determine the Rasmussen invariant of an almost positive knot in terms of its almost positive knot diagram. As corollaries, we prove that all almost positive knots are not homogeneous, and there is no almost positive knot of 4-genus one.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2014

Footnotes

This work was supported by JSPS KAKENHI Grant number 25001362.

References

[1] Abe, T., The Rasmussen invariant of a homogeneous knot. Proc. Amer. Math. Soc. 139 (2011, no. 7, 26472656.Google Scholar
[2] Baader, S., Quasipositivity and homogeneity. Math. Proc. Cambridge Philos. Soc. 139 (2005, no. 2, 287290.http://dx.doi.org/10.1017/S0305004105008698 Google Scholar
[3] Baader, S., Slice and Gordian numbers of track knots. Osaka J. Math. 42 (2005, no. 1, 257271.Google Scholar
[4] Banks, J. E., Homogeneous links, Seifert surfaces, digraphs and the reduced Alexander polynomial. Geom. Dedicata 166 (2013, 6798.http://dx.doi.org/10.1007/s10711-012-9786-1 Google Scholar
[5] Bar-Natan, D., On Khovanov's categorification of the Jones polynomial. Algebr. Geom. Topol. 2 (2002, 337370 (electronic).http://dx.doi.org/10.2140/agt.2002.2.337 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, 12401266.http://dx.doi.org/10.4153/CJM-2008-053-1 Google Scholar
[7] Cromwell, P. R., Homogeneous links. J. London Math. Soc. (2) 39 (1989, no. 3, 535552.http://dx.doi.org/10.1112/jlms/s2-39.3.535 Google Scholar
[8] González Manchón, P. M., Homogeneous links and the Seifert matrix. Pacific J. Math. 255 (2012, no. 2, 373392.http://dx.doi.org/10.2140/pjm.2012.255.373 Google Scholar
[9] Hirasawa, M., Triviality and splittability of special almost alternating links via canonical Seifert surfaces. Topology Appl. 102 (2000, no. 1, 89100.http://dx.doi.org/10.1016/S0166-8641(98)00138-2 Google Scholar
[10] Khovanov, M., A categorification of the Jones polynomial. Duke Math. J. 101 (2000, no. 3, 359426.http://dx.doi.org/10.1215/S0012-7094-00-10131-7 Google Scholar
[11] Nakamura, T., Four-genus and unknotting number of positive knots and links. Osaka J. Math. 37 (2000, no. 2, 441451.Google Scholar
[12] Przytycki, J. H., Positive knots have negative signature. Bull. Polish Acad. Sci. Math. 37 (1989, no. 712, 559562.Google Scholar
[13] Przytycki, J. H. and K. Taniyama, Almost positive links have negative signature. J. Knot Theory Ramifications 19 (2010, no. 2, 187289.http://dx.doi.org/10.1142/S0218216510007838 CrossRefGoogle Scholar
[14] Rasmussen, J., Khovanov homology and the slice genus. Invent. Math. 182 (2010, no. 2, 419447.http://dx.doi.org/10.1007/s00222-010-0275-6 Google Scholar
[15] Rudolph, L., Positive links are strongly quasipositive. In: Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr., 2, Geom. Topol. Publ., Coventry, 1999, pp. 555562.Google Scholar
[16] Shumakovitch, A. N., Rasmussen invariant, slice-Bennequin inequality, and sliceness of knots. J. Knot Theory Ramifications 16 (2007, no. 10, 14031412.http://dx.doi.org/10.1142/S0218216507005889 Google Scholar
[17] Stoimenow, A., Knots of genus two.arxiv:math/0303012v1.Google Scholar
[18] Stoimenow, A., On polynomials and surfaces of variously positive links. J. Eur. Math. Soc. (JEMS) 7 (2005, no. 4, 477509.http://dx.doi.org/10.4171/JEMS/36 Google Scholar
[19] Tagami, K., The maximal degree of the Khovanov homology of a cable link. Algebr. Geom. Topol. 13 (2013, no. 5, 28452896. http://dx.doi.org/10.2140/agt.2013.13.2845 Google Scholar