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Minimal genus and fibering of canonical surfaces via disk decomposition

Part of: Low-dimensional topology Symplectic geometry, contact geometry Topological manifolds

Published online by Cambridge University Press:  01 May 2014

A. Stoimenow
A. Stoimenow*
Affiliation:
Gwangju Institute of Science and Technology, School of General Studies, GIST College, 123 Cheomdan-gwagiro, 1 Oryong-dong, Buk-gu, Gwangju 500-712, Korea email stoimeno@stoimenov.nethttp://stoimenov.net/stoimeno/homepage/

Abstract

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This paper contains some applications of the description of knot diagrams by genus, and Gabai’s methods of disk decomposition. We show that there exists no genus one knot of canonical genus 2, and that canonical genus 2 fiber surfaces realize almost every Alexander polynomial only finitely many times (partially confirming a conjecture of Neuwirth).

MSC classification

Secondary: 57M25: Knots and links in $S^3$ 57N10: Topology of general $3$-manifolds 53D10: Contact manifolds, general 57M15: Relations with graph theory
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Issue 1 , 2014 , pp. 77 - 108
Copyright
© The Author 2014 

References

Adams, C. C., Das Knotenbuch (Spektrum Akademischer, Berlin, 1995) Translation of The Knot Book (W. H. Freeman & Co., New York 1994).Google Scholar
Alexander, J. W., ‘Topological invariants of knots and links’, Trans. Amer. Math. Soc. 30 (1928) 275–306.CrossRefGoogle Scholar
Bennequin, D., ‘Entrelacements et équations de Pfaff’, Soc. Math. de France, Astérisque 107–108 (1983) 87–161.Google Scholar
Brittenham, M., ‘Bounding canonical genus bounds volume’, Preprint, 1998, available at http://www. math.unl.edu/∼mbritten/personal/pprdescr.html.Google Scholar
Brittenham, M. and Jensen, J., ‘Canonical genus and the Whitehead doubles of pretzel knots’, Preprint, 2006, arXiv:0608765 [math.GT].Google Scholar
Burde, G. and Zieschang, H., Knots (de Gruyter, Berlin, 1986).Google Scholar
Conway, J. H., ‘On enumeration of knots and links’, Computational Problems in abstract algebra (ed. Leech, J.; Pergamon Press, 1969) 329–358.Google Scholar
Cromwell, P. R., Knots and links Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
Cromwell, P. R., ‘Homogeneous links’, J. London Math. Soc. (2) 39 (1989) 535–552.CrossRefGoogle Scholar
Cromwell, P. R. and Morton, H. R., ‘Positivity of knot polynomials on positive links’, J. Knot Theory Ramifications 1 (1992) 203–206.CrossRefGoogle Scholar
Crowell, R., ‘Genus of alternating link types’, Ann. of Math. (2) 69 (1959) 258–275.CrossRefGoogle Scholar
Dowker, C. H. and Thistlethwaite, M. B., ‘Classification of knot projections’, Topology Appl. 16 (1983) 19–31.CrossRefGoogle Scholar
Furmaniak, R. and Rankin, S., ‘Knotilus, a knot visualization interface’, http://srankin.math.uwo.ca/.Google Scholar
Gabai, D., ‘Foliations and genera of links’, Topology 23 (1984) 381–394.CrossRefGoogle Scholar
Gabai, D., ‘The Murasugi sum is a natural geometric operation’, Low-dimensional topology (San Francisco, Calif., 1981) , Contemporary Mathematics 20 (American Mathematical Society, Providence, RI, 1983) 131–143.CrossRefGoogle Scholar
Gabai, D., ‘The Murasugi sum is a natural geometric operation II’, Combinatorial methods in topology and algebraic geometry (Rochester, NY, 1982) , Contemporary Mathematics 44 (American Mathematical Society, Providence, RI, 1985) 93–100.CrossRefGoogle Scholar
Gabai, D., ‘Detecting fibred links in $S^3$ ’, Comment. Math. Helv. 61 (1986) 519–555.CrossRefGoogle Scholar
Gabai, D., ‘Foliations and the topology of 3-manifolds II’, J. Differential Geom. 26 (1987) 461–478.Google Scholar
Goda, H., Hirasawa, M. and Yamamoto, R., ‘Almost alternating diagrams and fibered links in $S^3$ ’, Proc. London Math. Soc. 83 (2001) 472–492.CrossRefGoogle Scholar
Hirasawa, M., ‘The flat genus of links’, Kobe J. Math. 12 (1995) 155–159.Google Scholar
Hoste, J. and Thistlethwaite, M., ‘KnotScape, a knot polynomial calculation and table access program’, http://www.math.utk.edu/∼morwen.Google Scholar
Jones, V. F. R., ‘Hecke algebra representations of braid groups and link polynomials’, Ann. of Math. (2) 126 (1987) 335–388.CrossRefGoogle Scholar
Lickorish, W. B. R. and Thistlethwaite, M. B., ‘Some links with non-trivial polynomials and their crossing numbers’, Comment. Math. Helv. 63 (1988) 527–539.CrossRefGoogle Scholar
Menasco, W. W., ‘Closed incompressible surfaces in alternating knot and link complements’, Topology 23 (1986) 37–44.CrossRefGoogle Scholar
Menasco, W. W. and Thistlethwaite, M. B., ‘The Tait flyping conjecture’, Bull. Amer. Math. Soc. 25 (1991) 403–412.CrossRefGoogle Scholar
Morton, H. R., ‘Seifert circles and knot polynomials’, Proc. Camb. Phil. Soc. 99 (1986) 107–109.CrossRefGoogle Scholar
Morton, H. R., ‘Infinitely many fibred knots having the same Alexander polynomial’, Topology 17 (1978) 101–104.CrossRefGoogle Scholar
Morton, H. R., ‘Fibred knots with a given Alexander polynomial’, Knots, braids and singularities, Plans-sur-Bex, 1982 , Monogr. Enseign. Math. 31 (1983) 205–222.Google Scholar
Murasugi, K., ‘Jones polynomial and classical conjectures in knot theory’, Topology 26 (1987) 187–194.CrossRefGoogle Scholar
Murasugi, K., ‘On the genus of the alternating knot’, J. Math. Soc. Japan 10 (1958) 94–105; 235–248.Google Scholar
Murasugi, K., ‘On a certain subgroup of the group of an alternating link’, Amer. J. Math. 85 (1963) 544–550.CrossRefGoogle Scholar
Murasugi, K. and Przytycki, J., ‘The skein polynomial of a planar star product of two links’, Math. Proc. Cambridge Philos. Soc. 106 (1989) 273–276.CrossRefGoogle Scholar
Nakamura, T., ‘On the crossing number of a 2-bridge knot and the canonical genus of its Whitehead doubles’, Osaka J. Math. 43 (2006) 609–623.Google Scholar
Nakamura, T., ‘Braidzel surfaces and the Alexander polynomial’, Proceedings of the Workshop Intelligence of Low Dimensional Topology (Osaka City University, 2004) 25–34.Google Scholar
Oertel, U., ‘Closed incompressible surfaces in complements of star links’, Pacific J. Math. 111 (1984) 209–230.CrossRefGoogle Scholar
Ozawa, M., ‘Closed incompressible surfaces in complements of positive knots’, Comment. Math. Helv. 77 (2002) 235–243.CrossRefGoogle Scholar
Polyak, M. and Viro, O., ‘Gauss diagram formulas for Vassiliev invariants’, Int. Math. Res. Notes 11 (1994) 445–454.CrossRefGoogle Scholar
Quach Hongler, C. V., ‘Polynôme d’Alexander des nœuds fibrés’, C. R. Acad. Sci. Paris Sér. A-B 289 (1979) A375–A377.Google Scholar
Quach Hongler, C. V. and Weber, C., ‘On the topological invariance of Murasugi special components of an alternating link’, Math. Proc. Cambridge Philos. Soc. 137 (2004) 95–108.CrossRefGoogle Scholar
Rolfsen, D., Knots and links (Publish or Perish, 1976).Google Scholar
Rudolph, L., ‘Positive links are strongly quasipositive’, Proceedings of the Kirbyfest, Geometry and Topology Monographs 2 (1999) 555–562; see also http://www.maths.warwick.ac.uk/gt/GTMon2/paper25.abs.html.CrossRefGoogle Scholar
Seifert, H., ‘Über das Geschlecht von Knoten’, Math. Ann. 110 (1934) 571–592.CrossRefGoogle Scholar
Silver, D. S., Stoimenow, A. and Williams, S. G., ‘Euclidean Mahler measure and Twisted Links’, Algebr. Geom. Topol. 6 (2006) 581–602.CrossRefGoogle Scholar
Stoimenow, A., ‘Knots of (canonical) genus two’, Fund. Math. 200 (2008) 1–67; arXiv:0303012 [math.GT].CrossRefGoogle Scholar
Stoimenow, A., ‘Knots of genus one’, Proc. Amer. Math. Soc. 129 (2001) 2141–2156.CrossRefGoogle Scholar
Stoimenow, A., ‘On the crossing number of semiadequate links’, Forum Math., published online (2012), doi:10.1515/forum-2011-0121.CrossRefGoogle Scholar
Stoimenow, A., ‘Gauß sum invariants, Vassiliev invariants and braiding sequences’, J. Knot Theory Ramifications 9 (2000) 221–269.CrossRefGoogle Scholar
Stoimenow, A., ‘Diagram genus, generators and applications’, Preprint, arXiv:1101.3390.Google Scholar
Stoimenow, A., ‘Realizing Alexander Polynomials by Hyperbolic Links’, Expo. Math. 28 (2010) 133–178.CrossRefGoogle Scholar
Stoimenow, A., ‘The crossing number and maximal bridge length of a knot diagram’, Pacific J. Math. 210 (2003) 189–199; with an appendix by M. Kidwell.Google Scholar
Stoimenow, A. and Vdovina, A., ‘Counting alternating knots by genus’, Math. Ann. 333 (2005) 1–27.CrossRefGoogle Scholar
Tripp, J. J., ‘The canonical genus of Whitehead doubles of a family torus knots’, J. Knot Theory Ramifications 11 (2002) 1233–1242.CrossRefGoogle Scholar
Yokota, Y., ‘Polynomial invariants of positive links’, Topology 31 (1992) 805–811.CrossRefGoogle Scholar
Zulli, L., ‘The rank of the trip matrix of a positive knot diagram’, J. Knot Theory Ramifications 6 (1997) 299–301.CrossRefGoogle Scholar