# Morse-Novikov Number

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 [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 S*^{3}, 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.