焊接纽结与带状环面纽结的非平凡性

Welded extensions and ribbon restrictions of diagrammatical moves

In this paper, we consider local moves on classical and welded diagrams of string links, and the notion of welded extension of a classical move. Such extensions being non-unique in general, the idea is to find a topological criterion which could isolate one extension from the others. To that end, we turn to the relation between welded string links and knotted surfaces in $\mathbb{R}^4$, and the ribbon sublclass of these surfaces. This provides the topological interpretation of classical local moves as surgeries on surfaces, and of welded local moves as surgeries on ribbon surfaces. Comparing t…

焊接弦链环与带状管的有限型不变量研究

On finite type invariants of welded string links and ribbon tubes

本文利用焊接对象研究带状纽结曲面的有限型不变量，分析w_k等价下焊接弦链环的性质，证明低阶时该等价关系可刻画有限型不变量信息，并探究其代数结构，结果对带状纽结曲面有直接推论。

Homotopy classification of ribbon tubes and welded string 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 le…

Virtual, Welded, and Ribbon Links in Arbitrary Dimensions

We define a generalization of virtual links to arbitrary dimensions by extending the geometric definition due to Carter et al. We show that many homotopy type invariants for classical links extend to invariants of virtual links. We also define generalizations of virtual link diagrams and Gauss codes to represent virtual links, and use such diagrams to construct a combinatorial biquandle invariant for virtual $2$-links. In the case of $2$-links, we also explore generalizations of Fox-Milnor movies to the virtual case. In addition, we discuss definitions extending the notion of welded links to h…

On the welded Tube map

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.

实心带状环面链环的Reidemeister定理

A Reidemeister Theorem for Solid Ribbon Torus Links

针对焊接链环的Reidemeister刻画这一长期未解问题，本文提出了实心带状环面链环（R4中仅含带状奇点的浸入实心环面）的Reidemeister定理，在广义带状同痕下给出完整刻画。