A Short Proof of a Theorem of Tanaka On Composite Knots with Symmetric Union Presentations

TL;DR

This paper provides a concise proof of Tanaka's theorem, establishing that certain symmetric union presentations of composite ribbon knots imply the presence of non-trivial knot summands and their mirrors.

Contribution

The paper offers a simplified proof of Tanaka's theorem regarding symmetric union presentations of composite ribbon knots.

Findings

Symmetric union presentation with one twisting region implies non-trivial knot summands.

The theorem applies to both a knot and its mirror image.

The proof is notably shorter than previous versions.

Abstract

We present a short proof of a theorem of Tanaka that if a composite ribbon knot admits a symmetric union presentation with one twisting region, then it has a non-trivial knot and its mirror image as connected summands.