Two more proofs that the Kinoshita graph is knotted

TL;DR

This paper provides two elementary proofs demonstrating that the Kinoshita graph, a complex embedding in the 3-sphere, is knotted, despite each edge removal resulting in an unknotted loop.

Contribution

It introduces two simple proofs using classical knot theory theorems to establish the knottedness of the Kinoshita graph, accessible to those with basic knot theory knowledge.

Findings

Kinoshita graph is knotted.

Removal of any edge yields an unknotted loop.

Two classical theorems suffice for the proofs.

Abstract

The Kinoshita graph is a particular embedding in the 3-sphere of a graph with three edges, two vertices and no loops. It has the remarkable property that although the removal of any edge results in an unknotted loop, the Kinoshita graph is itself knotted. We use two classical theorems from knot theory to give two particularly simple proofs that the Kinoshita graph is knotted. Apart from appealing to the two classical theorems, the exposition is elementary and is aimed at those with only a passing familiarity with knot theory.