ON THE WELDED TUBE MAP

Abstract

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 S 4. It emphasizes the fact that ribbon torus-knots with a given filling are in one-to-one correspondence with welded knots before quo-tient under classical Reidemeister moves and reformulates these moves and the known sources of non-injectivity of the Tube map in terms of filling changes. This note investigates the already known connection between welded knots, that is a quotient of the virtual knot theory, and ribbon torus-knots, that is a restricted notion of knotted surfaces in S 4. Virtual knots are a completion of usual knots thought of as abstract circles with a finite number of ori-ented and signed pairs of merged points. Indeed, every such pair describes a crossing, the rest of the circle describes how these crossings are connected and, up to isotopy, this data is sufficient to recontruct the knot in R3. However, not all pairings are realizable without introducing some additional crossing. Allowing non reported virtual crossings remedy this deficiency. Many knot invariants can be combinatorially ex-tended to the virtual case; in particular knot groups, which are the fundamental groups of the complement of knots in S 3, and all its derivative. This combinatorial notion of virtual knot group may disconcert since virtual knot theory does have a topological realization as knots in thickened surfaces modulo handle sta