Constructions of Morse maps for knots and links, and upper bounds on the Morse-Novikov number

Abstract

The Morse-Novikov number MN(L) of an oriented link L in the 3-sphere is the minimum number of critical points of a Morse map from the complement of L in the 3-sphere to the circle representing the class of a Seifert surface for L (e.g., the Morse-Novikov number of L is zero if and only if L is fibered). We develop various constructions of Morse maps (Milnor maps, Stallings twists, splicing along a link which is a closed braid with respect to a Morse map, Murasugi sums, cutting a Morse map along an arc on a page) and use them to bound Morse-Novikov numbers from above in terms of other knot and link invariants (free genus, crossing number, braid index, wrapping genus and layered wrapping genus).