Handle number is not always realized by a minimal genus Seifert surface

TL;DR

This paper constructs specific genus one knots where the handle number is only realized by non-minimal genus Seifert surfaces, challenging previous conjectures about the relationship between Seifert genus and Morse-Novikov genus.

Contribution

It provides counterexamples to the conjecture linking Seifert genus with minimal handle number, and introduces the genus g Morse-Novikov number to study these discrepancies.

Findings

Counterexamples to the conjecture relating Seifert genus and handle number

Additivity of Morse-Novikov genus and number under connected sum

Possibility of arbitrarily large differences between these invariants

Abstract

We construct genus one knots whose handle number is only realized by Seifert surfaces of non-minimal genus. These are counterexamples to the conjecture that the Seifert genus of a knot is its Morse-Novikov genus. As the Morse-Novikov genus may be greater than the Seifert genus, we define the genus $g$ Morse-Novikov number $MN_g(L)$ as the minimum handle number among Seifert surfaces for $L$ of genus $g$. Since, as we further show, the Morse-Novikov genus and the minimal genus Morse-Novikov number are additive under connected sum of knots, it then follows that there exists examples for which the discrepancies between Seifert genus and Morse-Novikov genus and between the Morse-Novikov number and the minimal genus Morse-Novikov number can be made arbitrarily large.