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42 (2005), 557–572. 6. M. Hirasawa and L. Rudolph, Constructions of Morse maps for knots and links, and upper bounds on the Morse-Novikov number, preprint. 7. T. Kanenobu, The augmentation subgroup of a pretzel link, Math. Sem. Notes Kobe Univ. 7 (1979), 363–384. 8. J. Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. ; University of Tokyo Press, Tokyo 1968. 9. A. Pajitnov, Closed orbits of gradient flows and logarithms of non-abelian Witt vectors, Special issues dedicated to Daniel Quillen on the occasion of his sixtieth birthday, Part V.

A Morse map f : CL → S 1 is said to be minimal if for each i the number mi (f ) is minimal on the class of all regular maps homotopic to f . Under these notations, the following basic theorem is shown ([10]). 1 ([10]). There is a minimal Morse map satisfying: (1) m0 (f ) = m3 (f ) = 0; March 4, 2007 11:41 WSPC - Proceedings Trim Size: 9in x 6in ws-procs9x6 36 (2) All critical values of the same index coincide; (3) f −1 (x) is a Seifert surface of L for any regular value x. 1 is said to be moderate.

See Fig. 2. ) be an arc properly embedded in M as illustrated in Fig. 2, and set M = M − IntN (α1 ∪ α2 ). Then we may March 4, 2007 11:41 WSPC - Proceedings Trim Size: 9in x 6in ws-procs9x6 39 regarded (M , γ) as a sutured manifold such that M is homeomorphic to D2 × S 1 . By the product decomposition as in Fig. 1, we have (M , γ) is a product sutured manifold. 1. Since it is known that L is not fibred, we have MN (L) = 2 × h(R) = 2. α1 R+(γ) γ γ R-(γ) s(γ) R-(γ) α2 Fig. 2. s(γ) D D s(γ) s(γ') M' M'' Fig.

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