Narrow Results

Satoh has defined a construction to obtain a ribbon torus knot given a welded knot. This construction is known to be surjective. We show that it is not injective. Using the invariant of the peripheral structure, it is possible to provide a restriction on this failure of injectivity. In particular we also provide an algebraic classification of the construction when restricted to classical knots, where it is equivalent to the torus spinning construction.

We describe a method of encoding various types of link diagrams, including those with classical, flat, rigid, welded, and virtual crossings. We show that this method may be used to encode link diagrams, up to equivalence, in a notation whose length is a cubic function of the number of "riser marks". For classical knots, the minimal number of such marks is twice the bridge index, and a classical knot diagram in minimal bridge form with bridge index b may be encoded in space . A set of moves on the notation is defined. As a demonstration of the utility of the notation we give another proof that the Kishino virtual knot is non-classical.

Using Gauss diagrams, one can define the virtual bridge number vb(K) and the welded bridge number wb(K), invariants of virtual and welded knots satisfying wb(K) ≤ vb(K). If K is a classical knot, Chernov and Manturov showed that vb(K) = br(K), the bridge number as a classical knot, and we ask whether the same thing is true for welded knots. The welded bridge number is bounded below by the meridional rank of the knot group G_K, and we use this to relate this question to a conjecture of Cappell and Shaneson. We show how to use other virtual and welded invariants to further investigate bridge numbers. Among them are Manturov's parity and the reduced virtual knot group Ḡ_K, and we apply these methods to address Questions 6.1–6.3 and 6.5 raised by Hirasawa, Kamada and Kamada in their paper [Bridge presentation of virtual knots, J. Knot Theory Ramifications 20(6) (2011) 881–893].

Spun-knots (respectively, spinning tori) in $S^4$ made from classical 1-knots compose an important class of 2-knots (respectively, embedded tori) contained in $S^4$. Virtual 1-knots are generalizations of classical 1-knots. We generalize these constructions to the virtual 1-knot case by using what we call, in this paper, the spinning construction of submanifolds. The construction proceeds as follows: For a virtual 1-knot $K$, take an embedded circle $C$ contained in (a closed oriented surface $F$)×(a closed interval $[0,1]$), where $F$ is called a representing surface in virtual 1-knot theory. Embed $F$ in $S^4$ by an embedding map $f$, and let $F$ stand for $f(F).$ Regard the tubular neighborhood of $F$ in $S^4$ as the result of rotating $F\times [0,1]$ around $F$. Rotate $C$ together then with $F\times [0,1]$. When $C\cap (F\times \{0\})=\varphi$, we obtain an embedded torus $Q\subset S^4$. We prove the following: The embedding type $Q$ in $S^4$ depends only on $K$, and does not depend on $f$. Furthermore, the submanifolds, $Q$ and “the embedded torus made from $K$ by using Satoh’s method”, of $S^4$ are isotopic.

Fiberwise equivalence of diagrams refers to fiberwise equivalence of tori in 4-space that lie over the diagrams. We prove that two virtual 1-knot diagrams $\alpha$ and $\beta$ are fiberwise equivalent if and only if $\alpha$ and $\beta$ are rotational welded equivalent (see the body of the paper for this definition).

We generalize the construction in the virtual 1-knot case written in the first paragraph, and we also succeed to make a consistent construction of one-dimensional-higher submanifolds from any virtual two-dimensional knot. Note that Satoh’s method says nothing about the virtual 2-knot case. Rourke’s interpretation of Satoh’s method is that one puts “fiber-circles” on each point of each virtual 1-knot diagram. If there is no virtual branch point in a virtual 2-knot diagram, our way gives such fiber-circles to each point of the virtual 2-knot diagram. Furthermore we prove the following: If a virtual 2-knot diagram $\alpha$ has a virtual branch point, $\alpha$ cannot be covered by such fiber-circles. Hence Rourke’s method cannot be generalized to the virtual 2-knot case. Only the spinning construction introduced in this paper works for now.

The notion of a virtual link is a generalization of a classical link. Alexander numbering is a numbering of $\mathbb{Z}$ to arcs of a classical link diagram which is due to a numbering to disks of a complement of a link diagram in $S^2$. Every classical link diagram admits Alexander numbering. A virtual link diagram corresponds to a link diagram in a closed oriented surface. Some virtual link diagrams do not admit any Alexander numbering. If a virtual link diagram admits Alexander numbering, we call it an almost classical virtual link diagram. In this paper, we construct a map from the set of virtual link diagrams to that of almost classical virtual link diagrams. It induces a map from the set of virtual links to that of almost classical virtual links. Using this map, we define a kind of Goeritz matrix of virtual link diagrams and introduce invariants of virtual links.

An unknotting operation is a local move such that any knot diagram can be transformed into a diagram of the trivial knot by a finite sequence of these operations plus some Reidemeister moves. It is known that for all $n\ge 2$ the $H(n)$-move is an unknotting operation for classical knots and links. In this paper, we extend the classical unknotting operation $H(n)$-move to virtual knots and links. Virtualization and forbidden move are well-known unknotting operations for virtual knots and links. We also show that virtualization and forbidden move can be realized by a finite sequence of generalized Reidemeister moves and $H(n)$-moves.