By I. Shariguin
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Since 2-bridge knots and links are strongly invertible, simultaneous change of orientation does not matter. However, for 2-component links, we must specify which of the two possible orientations is assigned. As a convention, we specify the orientation of 2-bridge link S(p, q) in the pillow case form as in Figure 2. Fig. 2. The orientation of 2-bridge links As the result, S(4, 3) (Figure 2 left) is a fibered link of genus 1, and S(4, 1) (Figure 2 right) is a non-fibered link of genus 0. This convention is consistent with the fact 3/4 has continued fraction [2, 2, 2], and 1/4 has [4].
Goda and A. Pajitnov, Twisted Novikov homology and Circle-valued Morse theory for knots and links, Osaka J. Math. 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.
See, for example, [4] and [9]. Let L be an oriented link in the 3-sphere S 3 . A Morse map f : CL := 3 S − L → S 1 is said to be regular if each component of L, say Li , has a neighborhood framed as S 1 × D2 such that (i) Li = S 1 × {0} (ii) f |S 1 ×(D2 −{0}) → S 1 is given by (x, y) → y/|y|. We denote by mi (f ) the number of the critical points of f of index i. 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 .