Additivity of Handle Number and Morse-Novikov Number of a-small Knots

TL;DR

This paper proves that for a-small knots, both handle number and Morse-Novikov number are additive under connected sum, using circular thin position techniques.

Contribution

It establishes the additivity of handle number and Morse-Novikov number specifically for a-small knots, a result not previously known.

Findings

Handle number is additive under connected sum of a-small knots

Morse-Novikov number is additive under connected sum of a-small knots

Uses circular thin position to prove these properties

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.