Volume 10, issue 2 (2010)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 25, 1 issue

Volume 24, 9 issues

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1472-2739 (online)
ISSN 1472-2747 (print)
Author Index
To Appear
 
Other MSP Journals
On the tunnel number and the Morse–Novikov number of knots

Andrei Pajitnov

Algebraic & Geometric Topology 10 (2010) 627–635
Bibliography
1 B Clark, The Heegaard genus of manifolds obtained by surgery on links and knots, Internat. J. Math. Math. Sci. 3 (1980) 583 MR582900
2 H Doll, A generalized bridge number for links in $3$–manifolds, Math. Ann. 294 (1992) 701 MR1190452
3 H Goda, On handle number of Seifert surfaces in $S^3$, Osaka J. Math. 30 (1993) 63 MR1200821
4 H Goda, Some estimates of the Morse–Novikov numbers for knots and links, from: "Intelligence of low dimensional topology 2006" (editors J S Carter, S Kamada, L H Kauffman, A Kawauchi, T Kohno), Ser. Knots Everything 40, World Sci. Publ., Hackensack, NJ (2007) 35 MR2371706
5 H Goda, A V Pajitnov, Twisted Novikov homology and circle-valued Morse theory for knots and links, Osaka J. Math. 42 (2005) 557 MR2166722
6 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 MR2488751
7 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 arXiv:math.GT/0311134
8 D Kim, J Lee, Some invariants of pretzel links, Bull. Austral. Math. Soc. 75 (2007) 253 MR2312569
9 T Kobayashi, A construction of arbitrarily high degeneration of tunnel numbers of knots under connected sum, J. Knot Theory Ramifications 3 (1994) 179 MR1279920
10 T Kobayashi, Y Rieck, On the growth rate of the tunnel number of knots, J. Reine Angew. Math. 592 (2006) 63 MR2222730
11 T Kohno, Tunnel numbers of knots and Jones–Witten invariants, from: "Braid group, knot theory and statistical mechanics, II" (editors C N Yang, M L Ge), Adv. Ser. Math. Phys. 17, World Sci. Publ. (1994) 275 MR1338606
12 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)
13 M Lustig, Y Moriah, Generalized Montesinos knots, tunnels and $\mathcal N$–torsion, Math. Ann. 295 (1993) 167 MR1198847
14 K Morimoto, On the additivity of tunnel number of knots, Topology Appl. 53 (1993) 37 MR1243869
15 K Morimoto, There are knots whose tunnel numbers go down under connected sum, Proc. Amer. Math. Soc. 123 (1995) 3527 MR1317043
16 K Morimoto, On the super additivity of tunnel number of knots, Math. Ann. 317 (2000) 489 MR1776114
17 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 MR1356163
18 K Morimoto, M Sakuma, Y Yokota, Identifying tunnel number one knots, J. Math. Soc. Japan 48 (1996) 667 MR1404816
19 S P Novikov, Multivalued functions and functionals. An analogue of the Morse theory, Dokl. Akad. Nauk SSSR 260 (1981) 31 MR630459
20 A V Pajitnov, On the Novikov complex for rational Morse forms, Ann. Fac. Sci. Toulouse Math. $(6)$ 4 (1995) 297 MR1344724
21 A V Pajitnov, Circle-valued Morse theory, de Gruyter Studies in Math. 32, de Gruyter (2006) MR2319639
22 L Rudolf, Murasugi sums of Morse maps to the circle, Morse–Novikov numbers, and free genus of knots arXiv:math.GT/0108006
23 M Scharlemann, J Schultens, The tunnel number of the sum of $n$ knots is at least $n$, Topology 38 (1999) 265 MR1660345
24 M Scharlemann, J Schultens, Annuli in generalized Heegaard splittings and degeneration of tunnel number, Math. Ann. 317 (2000) 783 MR1777119
25 H Schubert, Über eine numerische Knoteninvariante, Math. Z. 61 (1954) 245 MR0072483
26 K Veber, A V Pajitnov, L Rudolf, The Morse–Novikov number for knots and links, Algebra i Analiz 13 (2001) 105 MR1850189