Computing the Morse–Novikov Number of 2-Bridge Links

The Morse–Novikov number [Formula: see text] is the minimum number of critical points of a regular Morse map [Formula: see text] representing the canonical cohomology class. We combine Hoste’s lattice-walk formula for the multivariable Alexander polynomial with a supporting-diagonal criterion to compute [Formula: see text] for oriented 2-bridge links. For five infinite families, we determine fiberedness separately for the two orientation classes by testing the monicity of the corresponding specializations. For each orientation in these families, the Morse–Novikov number is [Formula: see text] in the fibered cases and [Formula: see text] otherwise; in many cases both orientations have value [Formula: see text]. In particular, we determine when reversing one component changes fiberedness within these families.