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).

arXiv comment23 figures; supercedes, and considerably extends, mathGT/0108006

References in corpus (2)

• Géométrie de contact: de la dimension trois vers les dimensions supérieures
• Algebraic functions and closed braids

Cited by in corpus (5)

• On the tunnel number and the Morse-Novikov number of knots
• Twisted Novikov homology and circle-valued Morse theory for knots and links
• Circular handle decompositions of free genus one knots
• Whitehead's Integral Formula, Isolated Critical Points, and the Enhancement of the Milnor Number
• Meromorphic functions, bifurcation sets and fibred links