A partial classification of 3-dimensional clasp number two, genus two fibered knots

TL;DR

This paper classifies certain genus two fibered knots with clasp number two, focusing on those admitting a specific type of clasp disk, thereby advancing understanding of their geometric properties.

Contribution

It provides a partial classification of genus two fibered knots with clasp number two, introducing the concept of clasp disks of type II.

Findings

Classification of clasp number two, genus two fibered knots

Introduction of clasp disks of type II

Conditions under which these knots admit such clasp disks

Abstract

The 3-dimensional clasp number $cl(K)$ of a knot $K$ is the minimum number of clasp singularities of clasp disk, a singular immersed disk bounding $K$ whose singular set consists of only clasp singularities. We give a classification of clasp number two, genus two fibered knots under the assumption that they admit a clasp disk of certain type which we call of type II.