The boundaries of dipole graphs and the complete bipartite graphs K_{2,n}

TL;DR

This paper explores the relationship between graph embeddings and Seifert surfaces of links, proving that any link can be bounded by a surface derived from a specific bipartite graph embedding, and provides an algorithm for constructing such surfaces.

Contribution

It establishes a novel connection between link boundaries and embeddings of complete bipartite graphs, and introduces an algorithm for constructing corresponding graph diagrams.

Findings

Every link can be realized as the boundary of a surface from a $K_{2,n}$ graph embedding.

An explicit algorithm for constructing graph diagrams of links is provided.

The algorithm is demonstrated on specific links $4_1^2$ and $5_2$.

Abstract

We study the Seifert surfaces of a link by relating the embeddings of graphs by using induced graphs. As applications, we prove that every link $L$ is the boundary of an oriented surface which is obtained from a graph embedding of a complete bipartite graph $K_{2,n}$, where all voltage assignments on the edges of $K_{2,n}$ are 0. We also provide an algorithm to construct such a graph diagram of a given link and demonstrate the algorithm by dealing with the links $4_1^2$ and $5_2$.