On the welded Tube map

TL;DR

This paper explores the relationship between welded knots and ribbon torus-knots via the Tube map, highlighting how filling changes relate to classical moves and non-injectivity issues.

Contribution

It reformulates the Tube map in terms of filling changes, providing new insights into the correspondence between welded knots and ribbon torus-knots.

Findings

Ribbon torus-knots with a fixed filling correspond to welded knots before quotienting.

Reformulation of Reidemeister moves in terms of filling changes.

Identification of sources of non-injectivity in the Tube map.

Abstract

This note investigates the so-called Tube map which connects welded knots, that is a quotient of the virtual knot theory, to ribbon torus-knots, that is a restricted notion of fillable knotted tori in the 4-sphere. It emphasizes the fact that ribbon torus-knots with a given filling are in one-to-one correspondence with welded knots before quotient under classical Reidemeister moves and reformulates these moves and the known sources of non-injectivity of the Tube map in terms of filling changes.