Narrow Results

For a signed cyclic graph $G$, we can construct a unique virtual link $L$ by taking the medial construction and converting 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In this paper, we shall prove that a virtual link $L$ is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial $F[G]$ for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between $F[G]$ of a signed cyclic graph $G$ and the bracket polynomial of one of the virtual link diagrams associated with $G$. Finally, we give a spanning subgraph expansion for $F[G]$.

An embedding presentation of a diagram is introduced, which has proved to be a unique presentation of a diagram. Let ${ℒ}$ be a set of all diagrams, called also links in this paper. An algebraic system $({ℒ},\sim )$ is constructed. In fact, a link in $R^3$ (or $S^3$) is the equivalent class $[L]$ where $L$ is one of its embedding presentations. Based on $({ℒ},\sim )$, Reduction Crossing Algorithm is proposed which is used to reduce the number of crossings in an embedding presentation by introducing a main tool called a pass replacement. For an infinite set of unknots ${𝒰}$, each $K$ in ${𝒰}$ can be transformed into the trivial unknot in at most $O(n^c)$ by applying the algorithm where $c$ is a constant, $K\in {𝒰}$ and $n=|V(K)|$. As special consequences, three unknots are unknotted, which are Goeritz’s unknot, Thistlethwaite’s unknot and Haken’s unknot (image courtesy of Cameron Gordon). Moreover, an infinite family of unknots $K_{G_{2k,2l}}\in {𝒰}$ are unknotted in $O(nloglogn)$ time. In addition, unique presentations of a virtual link, an oriented link and oriented virtual link are introduced, respectively.

We extend the theory of hyperbolicity of links in the 3-sphere to tg-hyperbolicity of virtual links, using the fact that the theory of virtual links can be translated into the theory of links living in closed orientable thickened surfaces. When the boundary surfaces are taken to be totally geodesic, we obtain a tg-hyperbolic structure with a unique associated volume. We prove that all virtual alternating links are tg-hyperbolic. We further extend tg-hyperbolicity to several classes of non-alternating virtual links. We then consider bounds on volumes of virtual links and include a table for volumes of the 116 nontrivial virtual knots of four or fewer crossings, all of which, with the exception of the trefoil knot, turn out to be tg-hyperbolic.

In this paper, we give two new criteria of detecting the checkerboard colorability of virtual links by using the odd writhe and the arrow polynomial of virtual links, respectively. As a result, 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.

Virtual index cocycle is the 1-cochain that counts virtual crossings in the arcs of a virtual link diagram. We show how this cocycle can be used to reformulate and unify some known invariants of virtual links.

Checkerboard framings are an extension of checkerboard colorings for virtual links. According to checkerboard framings, in 2017, Dye obtained an independent invariant of virtual links: the cut point number. Checkerboard framings and cut points can be used as a tool to extend other classical invariants to virtual links. We prove that one of the conjectures in Dye’s paper is correct. Moreover, we analyze the connection and difference between checkerboard framing obtained from virtual link diagram by adapting cut points and twisted link diagram obtained from virtual link diagram by introducing bars. By adjusting the normalized arrow polynomial of virtual links, we generalize it to twisted links. We show that it is an invariant for twisted link. Finally, we figure out three characteristics of the normalized arrow polynomial of a checkerboard colorable twisted link, which is a tool of detecting checkerboard colorability of a twisted link. The latter two characteristics are the same as in the case of checkerboard colorable virtual link diagram.

In this paper, we provide two new congruences of the generalized Alexander polynomial $Z_L$ for periodic virtual links $L$. We use the Yang–Baxter state model of $Z_L$ introduced by Kauffman and Radford.

The arrow polynomial is an invariant of framed oriented virtual links that generalizes the virtual Kauffman bracket. In this paper, we define the homological arrow polynomial, which generalizes the arrow polynomial to framed oriented virtual links with labeled components. The key observation is that, given a link in a thickened surface, the homology class of the link defines a functional on the surface’s skein module, and by applying it to the image of the link in the skein module this gives a virtual link invariant. We give a graphical calculus for the homological arrow polynomial by taking the usual diagrams for the Kauffman bracket and including labeled “whiskers” that record intersection numbers with each labeled component of the link. We use the homological arrow polynomial to study $(\mathbb{Z}/n\mathbb{Z})$-nullhomologous virtual links and checkerboard colorability, giving a new way to complete Imabeppu’s characterization of checkerboard colorability of virtual links with up to four crossings. We also prove a version of the Kauffman–Murasugi–Thistlethwaite theorem that the breadth of an evaluation of the homological arrow polynomial for an “h-reduced” diagram $D$ is $4(c(D)-g(D)+1)$.

In this paper, we introduce two new polynomial invariants $Q_D(t)$ and $A_D(t)$ for one-component virtual doodles. We will also show that these polynomial invariants are not invariants of flat virtual knots.

The core group of a classical link was introduced independently by Kelly in 1991 and Wada in 1992. It is a link invariant defined by a presentation involving the arcs and crossings of a diagram, related to Wirtinger’s presentation of the fundamental group of a link complement. Two close relatives of the core group are defined by presentations involving regions rather than arcs; one of them is related to Dehn’s presentation of a link group. The definitions are extended to virtual link diagrams and properties of the resulting invariants are discussed.

In this paper, we answer a question raised in “Peripheral elements in reduced Alexander modules” [J. Knot Theory Ramifications 31 (2022) 2250058]. We also correct a minor error in that paper.

We give a geometric description of welded links in the spirit of Kuperberg's description of virtual links: “What is a virtual link?” [8].