Morse-Novikov Number

The projection map of the boundary of a knot complement to S¹ to its meridian (which is constant on the longitude) extends to a circle-valued Morse function on the complement. The minimal number of critical points for such an extension is called the "Morse-Novikov number" of a knot, MN(K). MN(K) = 0 if and only if K is fibered. Since the Euler characteristic of the complement is 0, MN(K) is always even.

The Morse-Novikov number was introduced in [5], in which estimates on its value are developed along with some computations. Goda, in [2], announced the computation of the value of MN(K) for all prime knots of crossing number 10 or less. (The value is 2 for all such knots that are not fibered.) This result was based on his earlier paper, [1].

The references below include other papers on the subject.

References

[1] Goda, H., On handle number of Seifert surfaces in S³, Osaka J. Math. 30 (1993), 63-80.

[2] Goda, H., 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) 3542.

[3] Goda, H. and Pajitnov, A., Twisted Novikov homology and circle-valued Morse thoery for knots and links, Osaka J. Math. 42 (2005), 557-572.

[4] Hirasawa, M. and Rudolph, L., Constructions of Morse maps for knots and links, and upper bounds on the Morse-Novikov number, ArXiv preprint.

[5] Pazhitnov, A., Rudolph, L., and Weber, L. K., The Morse-Novikov number for knots and links, (Russian) Algebra i Analiz 13 (2001), no. 3, 105--118; translation in St. Petersburg Math. J. 13 (2002), no. 3, 417--426.