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Register a new userOn the welded Tube map
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Authors
Benjamin Audoux
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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.
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Let $n$ be a positive integer. The aim of this paper is to study two local moves $V(n)$ and $V^{n}$ on welded links, which are generalizations of the crossing virtualization. We show that the $V(n)$-move is an unknotting operation on welded knots for any $n$, and give a classification of welded links up to $V(n)$-moves. On the other hand, we give a necessary condition for which two welded links are equivalent up to $V^{n}$-moves. This leads to show that the $V^{n}$-move is not an unknotting operation on welded knots except for $n=1$. We also discuss relations among $V^{n}$-moves, associated core groups and the multiplexing of crossings.
We prove that the crossing changes, Delta moves, and sharp moves are unknotting operations on welded knots.
For a classical link, Milnor defined a family of isotopy invariants, called Milnor $overline{mu}$-invariants. Recently, Chrisman extended Milnor $overline{mu}$-invariants to welded links by a topological approach. The aim of this paper is to show that Milnor $overline{mu}$-invariants can be extended to welded links by a combinatorial approach. The proof contains an alternative proof for the invariance of the original $overline{mu}$-invariants of classical links.
Ribbon 2-knotted objects are locally flat embeddings of surfaces in 4-space which bound immersed 3-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper we consider ribbon tubes and ribbon torus-links, which are natural analogues of string links and links, respectively. We show how ribbon tubes naturally act on the reduced free group, and how this action classifies ribbon tubes up to link-homotopy, that is when allowing each component to cross itself. At the combinatorial level, this provides a classification of welded string links up to self-virtualization. This generalizes a result of Habegger and Lin on usual string links, and the above-mentioned action on the reduced free group can be refined to a general virtual extension of Milnor invariants. As an application, we obtain a classification of ribbon torus-links up to link-homotopy.
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