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We introduce a new equivalence relation on decorated ribbon graphs, and show that its equivalence classes directly correspond to virtual links. We demonstrate how this correspondence can be used to convert any invariant of virtual links into an invariant of ribbon graphs, and vice versa.
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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.
A heap is a set with a certain ternary operation that is self-distributive (TSD) and exemplified by a group with the operation $(x,y,z)mapsto xy^{-1}z$. We introduce and investigate framed link invariants using heaps. In analogy with the knot group, we define the fundamental heap of framed links using group presentations. The fundamental heap is determined for some classes of links such as certain families of torus and pretzel links. We show that for these families of links there exist epimorphisms from fundamental heaps to Vinberg and Coxeter groups, implying that corresponding groups are infinite. A relation to the Wirtinger presentation is also described. The cocycle invariant is defined using ternary self-distributive (TSD) cohomology, by means of a state sum that uses ternary heap $2$-cocycles as weights. It is shown that the cohomology splits into two types, called degenerate and nondegenerate, and that the degenerate part is one dimensional. Subcomplexes are constructed based on group cosets, that allow computations of the nondegenerate part. We apply colorings inferred from fundamental heaps to compute cocycle invariants, and prove that the invariant values can be used to derive algebraic properties of the cohomology.
Multicrossings, which have previously been defined for classical knots and links, are extended to virtual knots and links. In particular, petal diagrams are shown to exist for all virtual knots.
In this paper we give two new criteria of detecting the checkerboard colorability of virtual links by using odd writhe and arrow polynomial of virtual links, respectively. By applying new criteria, we prove that 6 virtual knots are not checkerboard colorable, leaving only one virtual knot whose checkerboard colorability is unknown among all virtual knots up to four classical crossings.
Oikawa defined an unknotting operation on virtual knots, called a CF-move, and gave a classification of 2-component virtual links up to CF-moves by the virtual linking number and his $n$-invariant. In particular, it was proved that a CF-move characterizes the information contained in the virtual linking number for 2-component odd virtual links. In this paper, we extend this result by classifying odd virtual links and almost odd virtual links with arbitrary number of components up to CF-moves, using the virtual linking number. Moreover, we extend Oikawas $n$-invariant and introduce two invariants for 3-component even virtual links. Using these invariants together with the virtual linking number, we classify 3-component even virtual links up to CF-moves. As a result, a classification of 3-component virtual links up to CF-moves is provided.
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