The projection map of the boundary of a knot complement to S
The Morse-Novikov number was introduced in , in which estimates on its value are developed along with some computations. Goda, in , 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, .
The references below include other papers on the subject.
 Goda, H., On handle number of Seifert surfaces in S
 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.
 Goda, H. and Pajitnov, A., Twisted Novikov homology and circle-valued Morse thoery for knots and links, Osaka J. Math. 42 (2005), 557-572.
 Hirasawa, M. and Rudolph, L., Constructions of Morse maps for knots and links, and upper bounds on the Morse-Novikov number, ArXiv preprint.
 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.