An elementary proof for that all unoriented spanning surfaces of a link are related by attaching/deleting tubes and M\"{o}bius bands

TL;DR

This paper provides an elementary proof that all unoriented spanning surfaces of a link in 3-sphere are related through simple topological modifications like attaching or deleting tubes and Möbius bands.

Contribution

It offers a straightforward, elementary proof of a known result about the relationship between unoriented spanning surfaces of links.

Findings

All unoriented spanning surfaces are related by attaching/deleting tubes and Möbius bands.

The proof simplifies understanding of the topology of link surfaces.

Supports the classification of link surfaces through elementary operations.

Abstract

Gordon and Litherland showed that all compact, unoriented, possibly non-orientable surfaces in $S^3$ bounded by a link are realted by attaching/deleting tubes and half twisted bands. In this note we give an elementary proof for this result.