An equivalence between the set of graph-knots and the set of homotopy classes of looped graphs

TL;DR

This paper establishes a one-to-one correspondence between graph-knots and homotopy classes of looped graphs, providing a new way to relate these two mathematical structures through a simple formula.

Contribution

It introduces a novel bijective relationship between graph-knots and homotopy classes of looped graphs, linking these concepts explicitly.

Findings

Constructs a one-to-one correspondence between graph-knots and homotopy classes of looped graphs.

Shows the relationship between a graph-knot and the homotopy class derived from it.

Provides a simple formula to establish this correspondence.

Abstract

In the present paper we construct a one-to-one correspondence between the set of graph-knots and the set of homotopy classes of looped graphs. Moreover, the graph-knot and the homotopy class constructed from a given knot are related with this correspondence. This correspondence is given by a simple formula.