The Classification of Spun Torus Knots

TL;DR

This paper investigates the properties of a construction linking welded knots to ribbon torus knots, revealing its non-injective nature and providing an algebraic classification for classical knots.

Contribution

It demonstrates the non-injectivity of Satoh's construction and offers an algebraic classification when restricted to classical knots.

Findings

The construction is surjective but not injective.

Peripheral structure invariant restricts the failure of injectivity.

Classifies the construction algebraically for classical knots.

Abstract

S. Satoh has defined a construction to obtain a ribbon torus knot given a welded knot. This construction is known to be surjective. We show that it is not injective. Using the invariant of the peripheral structure, it is possible to provide a restriction on this failure of injectivity. In particular we also provide an algebraic classification of the construction when restricted to classical knots, where it is equivalent to the torus spinning construction.