On composite types of tunnel number two knots

TL;DR

This paper classifies the summands of composite tunnel number two knots using (g, b)-decompositions, providing a comprehensive table of their types based on these decompositions.

Contribution

It offers a complete classification of the connected sum components of tunnel number two knots through (g, b)-decomposition analysis.

Findings

Complete table of summands for composite tunnel number two knots.

Classification of knots into (3,0), (2,1), (1,2), or (0,3) types.

Analysis of connected sum decompositions based on (g, b)-decompositions.

Abstract

Let $K$ be a tunnel number two knot. Then, by considering the $(g, b)$-decompositions, $K$ is one of (3, 0)-, (2, 1)-, (1, 2)- or (0, 3)-knots. In the present paper, we analyze the connected sum summands of composite tunnel number two knots and give a complete table of those summands from the point of view of $(g, b)$-decompositions.