Classification of genus two knots which admit a (1,1) decomposition

TL;DR

This paper classifies genus two knots that can be decomposed into a genus one, one bridge position, detailing several families including bandings of genus one and cross cap knots, and certain satellite knots.

Contribution

It provides a comprehensive classification of genus two knots admitting a (1,1) decomposition, identifying specific families and their constructions.

Findings

Identifies families of genus two knots with (1,1) decompositions

Describes banding constructions involving genus one and cross cap knots

Includes classification of genus two tunnel number one satellite knots

Abstract

We describe the genus two knots which admit a genus one, one bridge position. These are divided into several families, one consists of vertical bandings of two genus one $(1,1)$-knots, other consists of vertical bandings of two cross cap number two 2-bridge knots, and the last one consists of genus two tunnel number one satellite knots.