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ESSENTIAL STATE SURFACES FOR KNOTS AND LINKS

Part of: Low-dimensional topology

Published online by Cambridge University Press:  19 March 2012

MAKOTO OZAWA
MAKOTO OZAWA*
Affiliation:
Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa University, 1-23-1 Komazawa, Setagaya-ku, Tokyo, 154-8525, Japan (email: w3c@komazawa-u.ac.jp)
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Abstract

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We study a canonical spanning surface obtained from a knot or link diagram, depending on a given Kauffman state. We give a sufficient condition for the surface to be essential. By using the essential surface, we can deduce the triviality and splittability of a knot or link from its diagrams. This has been done on the extended knot or link class that includes all semiadequate, homogeneous knots and links, and most algebraic knots and links. In order to prove the main theorem, we extend Gabai’s Murasugi sum theorem to the case of nonorientable spanning surfaces.

Keywords

state surfaceknot diagramadequate knothomogeneous knotMurasugi sum

MSC classification

Secondary: 57M25: Knots and links in $S^3$
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 91 , Issue 3 , December 2011 , pp. 391 - 404
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Adams, C., The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots (American Mathematical Society, Providence, RI, 2004), Revised reprint of the 1994 original.Google Scholar
[2]Aumann, R. J., ‘Asphericity of alternating knots’, Ann. of Math. (2) 64 (1956), 374–392.CrossRefGoogle Scholar
[3]Bonahon, F. and Siebenmann, L., ‘Geometric splittings of classical knots and the algebraic knots of Conway’, in: New Geometric Splittings of Classical Knots, London Mathematical Society Lecture Note Series, 75 (ed. Siebenmann, L.) (Cambridge University Press, Cambridge, 2003).Google Scholar
[4]Cromwell, P. R., ‘Homogeneous links’, J. Lond. Math. Soc. (2) 39 (1989), 535–552.CrossRefGoogle Scholar
[5]Cromwell, P. R., Knots and Links (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
[6]Curtis, C. L. and Taylor, S., ‘The Jones polynomial and boundary slopes of alternating knots’, arXiv:0910.4912.Google Scholar
[7]Diestel, R., Graph Theory, 3rd edn Graduate Texts in Mathematics, 173 (Springer, Berlin, 2005).Google Scholar
[8]Elliott, A., ‘State cycles, quasipositive modification, and constructing H-thick knots in Khovanov homology’, arXiv:0901.4039.Google Scholar
[9]Frankl, P. and Pontrjagin, L., ‘Ein Knotensatz mit Anwendung auf die Dimensionstheorie’, Math. Ann. 102 (1930), 785–789.CrossRefGoogle Scholar
[10]Futer, D., Kalfagianni, E. and Purcell, J. S., ‘Slopes and colored Jones polynomials of adequate knots’, Proc. Amer. Math. Soc. 139 (2011), 1889–1896.CrossRefGoogle Scholar
[11]Gabai, D., ‘The Murasugi sum is a natural geometric operation’, Contemp. Math. 20 (1983), 131–143.CrossRefGoogle Scholar
[12]Gabai, D., ‘Genera of the arborescent links’, Mem. Amer. Math. Soc. 59 (1986).Google Scholar
[13]Hatcher, A. and Thurston, W., ‘Incompressible surfaces in 2-bridge knot complements’, Invent. Math. 79 (1985), 225–246.CrossRefGoogle Scholar
[14]Hoste, J. and Thistlethwaite, M., ‘The Hoste-Thistlethwaite Table of 11 Crossing Knots’, in http://katlas.math.toronto.edu/wiki/The_Hoste-Thistlethwaite_Table_of_11_Crossing_Knots.Google Scholar
[15]Kauffman, L. H., ‘State models and the Jones polynomial’, Topology 26 (1987), 395–407.CrossRefGoogle Scholar
[16]Lickorish, W. B. R. and Thistlethwaite, M. B., ‘Some links with nontrivial polynomials and their crossing numbers’, Comment. Math. Helv. 63 (1988), 527–539.CrossRefGoogle Scholar
[17]Matveev, S., Algorithmic Topology and Classification of 3-manifolds, 2nd edn Algorithms and Computation in Mathematics, 9 (Springer, Berlin, 2007).Google Scholar
[18]Menasco, W., ‘Closed incompressible surfaces in alternating knot and link complements’, Topology 23 (1984), 37–44.CrossRefGoogle Scholar
[19]Menasco, W. and Thistlethwaite, M., ‘A geometric proof that alternating knots are nontrivial’, Math. Proc. Camb. Phil. Soc. 109 (1991), 425–431.CrossRefGoogle Scholar
[20]Menasco, W. and Thistlethwaite, M., ‘The classification of alternating links’, Ann. of Math. (2) 138 (1993), 113–171.CrossRefGoogle Scholar
[21]Neuwirth, L., ‘Interpolating manifolds for knots in S 3’, Topology 2 (1964), 359–365.CrossRefGoogle Scholar
[22]Ozawa, M., ‘Closed incompressible surfaces in the complements of positive knots’, Comment. Math. Helv. 77 (2002), 235–243.CrossRefGoogle Scholar
[23]Ozawa, M., ‘Nontriviality of generalized alternating knots’, J. Knot Theory Ramifications 15 (2006), 351–360.CrossRefGoogle Scholar
[24]Ozawa, M., ‘Rational structure on algebraic tangles and closed incompressible surfaces in the complements of algebraically alternating knots and links’, Topology Appl. 157 (2010), 1937–1948.CrossRefGoogle Scholar
[25]Ozawa, M. and Tsutsumi, Y., ‘Totally knotted Seifert surfaces with accidental peripherals’, Proc. Amer. Math. Soc. 131 (2003), 3945–3954.CrossRefGoogle Scholar
[26]Pabiniak, M. D., Przytycki, J. H. and Sazdanovic, R., ‘On the first group of the chromatic cohomology of graphs’, Geom. Dedicata 140 (2009), 19–48.CrossRefGoogle Scholar
[27]Rolfsen, D., Knots and Links (AMS Chelsea Publishing, Providence, RI, 2003).Google Scholar
[28]Seifert, H., ‘Über das Geschlecht von Knoten’, Math. Ann. 110 (1934), 571–592.CrossRefGoogle Scholar
[29]Thistlethwaite, M. B., ‘On the Kauffman polynomial of an adequate link’, Invent. Math. 93 (1998), 285–296.CrossRefGoogle Scholar