The Morse-Novikov Number of a Knot

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). Clearly, 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 a paper by Pajitnov, Rudolph, and Weber, referenced below, in which estimates on its value are developed along with some computations. Goda, in his paper on the subject, Some Estimates.., 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, On the handle number ... .

The references below include other papers on the subject.

References

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

H. Goda, Some estimates of the Morse-Novikov numbers for knots and links , proceedings of the conference ILDT 2006, preprint.

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

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

A. Pazhitnov, L. Rudolph, and L. K. Weber, 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.