Algebraic & Geometric Topology

On the tunnel number and the Morse–Novikov number of knots

Andrei Pajitnov

Full-text: Open access

Abstract

Let L be a link in S3; denote by N(L) the Morse–Novikov number of L and by t(L) the tunnel number of L. We prove that N(L)2t(L) and deduce several corollaries.

Article information

Source
Algebr. Geom. Topol., Volume 10, Number 2 (2010), 627-635.

Dates
Received: 19 October 2009
Accepted: 5 January 2010
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715108

Digital Object Identifier
doi:10.2140/agt.2010.10.627

Mathematical Reviews number (MathSciNet)
MR2606794

Zentralblatt MATH identifier
1196.57008

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds 57R35: Differentiable mappings 57R70: Critical points and critical submanifolds
Secondary: 57R19: Algebraic topology on manifolds 57R45: Singularities of differentiable mappings

Keywords
tunnel number Morse–Novikov number Alexander polynomial

Citation

Pajitnov, Andrei. On the tunnel number and the Morse–Novikov number of knots. Algebr. Geom. Topol. 10 (2010), no. 2, 627--635. doi:10.2140/agt.2010.10.627. https://projecteuclid.org/euclid.agt/1513715108


Export citation

References

  • B Clark, The Heegaard genus of manifolds obtained by surgery on links and knots, Internat. J. Math. Math. Sci. 3 (1980) 583–589
  • H Doll, A generalized bridge number for links in $3$–manifolds, Math. Ann. 294 (1992) 701–717
  • H Goda, On handle number of Seifert surfaces in $S\sp 3$, Osaka J. Math. 30 (1993) 63–80
  • H Goda, Some estimates of the Morse–Novikov numbers for knots and links, from: “Intelligence of low dimensional topology 2006”, (J S Carter, S Kamada, L H Kauffman, A Kawauchi, T Kohno, editors), Ser. Knots Everything 40, World Sci. Publ., Hackensack, NJ (2007) 35–42
  • H Goda, A V Pajitnov, Twisted Novikov homology and circle-valued Morse theory for knots and links, Osaka J. Math. 42 (2005) 557–572
  • H Goda, A V Pajitnov, Dynamics of gradient flows in the half-transversal Morse theory, Proc. Japan Acad. Ser. A Math. Sci. 85 (2009) 6–10
  • M Hirasawa, L Rudolph, Constructions of Morse maps for knots and links, and upper bounds on the Morse–Novikov number, to appear in J. Knot Theory Ramifications
  • D Kim, J Lee, Some invariants of pretzel links, Bull. Austral. Math. Soc. 75 (2007) 253–271
  • T Kobayashi, A construction of arbitrarily high degeneration of tunnel numbers of knots under connected sum, J. Knot Theory Ramifications 3 (1994) 179–186
  • T Kobayashi, Y Rieck, On the growth rate of the tunnel number of knots, J. Reine Angew. Math. 592 (2006) 63–78
  • T Kohno, Tunnel numbers of knots and Jones–Witten invariants, from: “Braid group, knot theory and statistical mechanics, II”, (C N Yang, M L Ge, editors), Adv. Ser. Math. Phys. 17, World Sci. Publ., River Edge, NJ (1994) 275–293
  • J H Lee, An upper bound for tunnel number of a knot using free genus, Lecture notes, $4$–th East Asian School of knots (2008) Available at \setbox0\makeatletter\@url http://faculty.ms.u-tokyo.ac.jp/~topology/EAS4slides/JungHoonLee.pdf {\unhbox0
  • M Lustig, Y Moriah, Generalized Montesinos knots, tunnels and $\mathcal N$–torsion, Math. Ann. 295 (1993) 167–189
  • K Morimoto, On the additivity of tunnel number of knots, Topology Appl. 53 (1993) 37–66
  • K Morimoto, There are knots whose tunnel numbers go down under connected sum, Proc. Amer. Math. Soc. 123 (1995) 3527–3532
  • K Morimoto, On the super additivity of tunnel number of knots, Math. Ann. 317 (2000) 489–508
  • K Morimoto, M Sakuma, Y Yokota, Examples of tunnel number one knots which have the property “$1+1=3$”, Math. Proc. Cambridge Philos. Soc. 119 (1996) 113–118
  • K Morimoto, M Sakuma, Y Yokota, Identifying tunnel number one knots, J. Math. Soc. Japan 48 (1996) 667–688
  • S P Novikov, Multivalued functions and functionals. An analogue of the Morse theory, Dokl. Akad. Nauk SSSR 260 (1981) 31–35
  • A V Pajitnov, On the Novikov complex for rational Morse forms, Ann. Fac. Sci. Toulouse Math. $(6)$ 4 (1995) 297–338
  • A V Pajitnov, Circle-valued Morse theory, de Gruyter Studies in Math. 32, de Gruyter, Berlin (2006)
  • L Rudolf, Murasugi sums of Morse maps to the circle, Morse–Novikov numbers, and free genus of knots
  • M Scharlemann, J Schultens, The tunnel number of the sum of $n$ knots is at least $n$, Topology 38 (1999) 265–270
  • M Scharlemann, J Schultens, Annuli in generalized Heegaard splittings and degeneration of tunnel number, Math. Ann. 317 (2000) 783–820
  • H Schubert, Über eine numerische Knoteninvariante, Math. Z. 61 (1954) 245–288
  • K Veber, A V Pajitnov, L Rudolf, The Morse–Novikov number for knots and links, Algebra i Analiz 13 (2001) 105–118