Additivity of handle number and Morse–Novikov number of a-small knots

Abstract

A knot is an a-small knot if its exterior does not contain closed incompressible surfaces disjoint from some incompressible Seifert surface for the knot. Using circular thin position for knots we prove that the handle number is additive under the connected sum of two a-small knots. As a consequence the Morse-Novikov number turns out to be additive under the connected sum of two a-small knots.

The Morse–Novikov number of knots under connected sum and cabling

“…In preparation for [12, Theorem 1.1], Manjarrez-Gutiérrez shows that the handle number of an a-small knot is realized by the handle number of an incompressible Seifert surface [12,Theorem 4.3]. On our way to Theorem 1.1 we prove the analogous Lemma 1.2 which removes the a-small hypothesis.…”

“…We refer the reader to [2,[6][7][8] for the basic elements of our approach to compression bodies and Heegaard splittings and to [3,11,18,21,22] for the core ideas of generalized Heegaard splittings, circular Heegaard splittings and circular generalized Heegaard splittings. For our purposes in this article, we take care to clarify the operations of weak reductions and amalgamations and refer the reader to [10,12,23,24] for further discussions of these operations.…”

“…This approach is rooted in Goda's work on handle numbers of sutured manifolds and Seifert surfaces [6,7] and Manjarrez-Gutiérrez's work on generalized circular Heegaard splittings and circular thin decompositions [11]. Notably, in [12,Theorem 1.1] Manjarrez-Gutiérrez uses this approach to prove Theorem 1.1 for the special class of a-small knots (that is, knots with no closed essential surface disjoint from a Seifert surface) using a key proposition about the positioning of the summing annulus for a connected sum with respect to a circular (locally) thin decompositions of the knot exterior which she established with Eudave-Muñoz [3,Proposition 5.1]. In essence, we manage to avoid this a-small hypothesis in our proof of Theorem 1.1 by paying closer attention to the behavior of counts of handles in generalized circular Heegaard splittings under the operations of weak reductions and amalgamations.…”

Knots and surfaces

“…For the handle number and the Morse-Novikov number, it holds that M N (K) = 2h(K) ( [56]). The handle number and the Morse-Novikov number are additive with respect to the connected sum of almost small knots ( [115]). It also holds that M N (K) ≤ 2t(K) ( [160]).…”

On non almost-fibered knots

“…Until now, the only known examples of non almost-fibered knots are some connected sum of knots, for instance, the connected sum of asmall knots [4]. In 2017, at the Mathematical Congress of Americas, Hans Boden asked whether there are non almost-fibered, hyperbolic knots.…”