The projection map of the boundary of a knot complement to S^{1} 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 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 S ^{3}*, 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.