Writhe polynomials and shell moves for virtual knots and links
In this paper, Nakamura, Nakanishi, and Satoh derive two results for virtual knots. The first result is that oriented virtual knots have the same writhe polynomial if and only if they are related by a set of moves which they call \textit{shell moves}. The second result is to show that two-component virtual links are classified up to shell moves in multiple ways using invariants of virtual links. The writhe polynomial of a virtual knot $K$, $W_K (t)$, was introduced by \textit{Z. Cheng} and \textit{H. Gao} in [J. Knot Theory Ramifications 22, No. 12, Article ID 1341002, 33 p. (2013; Zbl 1284.57005)]. It is given by the formula \[ W_K (t) = \sum_{n\neq 0} J_n (K)t^n-\sum_{n\neq 0} J_n (K) \] where $J_n(K)$ is the $n$-writhe of $K$. For more discussion of the $n$-writhe, see [\textit{S. Satoh} and \textit{K. Taniguchi}, Fundam. Math. 225, 327--341 (2014; Zbl 1302.57033)]. The authors introduce the notion of \textit{shell moves} for virtual knots and links. These are defined in terms of changes to the Gauss diagrams of the virtual links. The moves are illustrated in the following figure: {}\vspace{0.5cm} \begin{tikzpicture}[line width=1pt, >=Stealth] \draw [->] (0,-1) -- (0,1); \draw (0, 0) -- (1,0); \draw [<-] (1,-1) -- (1,1); \draw (0mm,4mm) arc [start angle=90, end angle=-90, radius=4mm]; \end{tikzpicture} \(\quad \overset{\mathrm{S}1}{\boldsymbol\sim} \quad\) \begin{tikzpicture}[line width=1pt, >=Stealth] \draw [->] (0,-1) -- (0,1); \draw (0, 0) -- (1,0); \draw [<-] (1,-1) -- (1,1); \draw (10mm,4mm) arc [start angle=-90, end angle=90, radius=-4mm]; \end{tikzpicture} \({}\qquad \qquad{}\) \begin{tikzpicture}[line width=1pt, >=Stealth] \draw [->] (0.0,0.0) -- (1.4,0.0); \draw (0.4,0.0) -- (0.4,1.0); \draw (1.0,0.0) -- (1.0,1.0); \end{tikzpicture} \(\quad \overset{\mathrm{S}2}{\boldsymbol\sim} \quad\) \begin{tikzpicture}[line width=1pt, >=Stealth] \draw [->] (0.0,0.0) -- (2.0,0.0); \draw (0.7cm,0.0cm) arc [start angle=0, end angle=180, radius=0.25cm]; \draw (1.6cm,0.0cm) arc [start angle=0, end angle=180, radius=0.25cm]; \draw (0.45cm,0.0cm) arc[start angle=180, end angle=115, radius=1cm]; \draw (1.55,1.8) arc[start angle=0, end angle=-65, radius=1cm]; \draw (1.35cm,0.0cm) arc[start angle=0, end angle=65, radius=1cm]; \draw (0.2,1.8) arc[start angle=180, end angle=245, radius=1cm]; \end{tikzpicture} \vspace{0.5 cm} The first main result in this paper is Theorem 1. The writhe polynomial is a complete invariant for virtual knots up to shell moves (and virtual isotopy). The second main result is somewhat more technical, and we suggest the interested reader consult the full paper for details. In essence, it states that there are multiple collections of complete invariants for two-component virtual links up to shell moves which can be constructed using the odd writhe, the linking number, and some other easily computed invariants defined in the paper. The main tool by which the first of these results is obtained is the following result: Lemma 2. Any Gauss diagram for a virtual knot is shell equivalent, up to virtual isotopy, to a diagram which is a concatenation of \textit{snails}. A \textit{snail} is a portion of a Gauss diagram consisting of half-circles with a single chord whose base is inside them, and whose head lies outside them; the rigorous definition and illustrations are given in Definition 2.5 and Figure 10 of the paper. This lemma is in itself a very interesting result about the shell equivalence of virtual knots. The lemma also allows the study of virtual knots up to shell moves to be reduced to the study of diagrams consisting of concatenations of snails. The results for two-component virtual links are proved by using a similar method: first, it is shown that any Gauss diagram of a two-component virtual link $L$ is shell equivalent, up to virtual isotopy, to a Gauss diagram built out of snails. This simplifies the problem of classifying two-component virtual links up to shell moves to the problem of classifying diagrams built out of snails, although this is somewhat more involved than classifying diagrams for virtual knots (one component links). Sections 4 and 5 of the paper are devoted to giving necessary and sufficient conditions for classifying these diagrams built of snails, and thus, for classifying virtual links with two components up to shell equivalence. The final section of the paper explores the relationships between the invariants used in the paper, and also shows that virtual links can be constructed to have specified values for many of the invariants. The results in this paper give a nice picture of the power of the writhe polynomial. Since polynomial invariants are easy to compare, finding strong polynomial invariants is an important goal in the study of links and virtual links. The paper shows exactly how powerful the odd writhe polynomial is for virtual knots, and also sheds light on the power of certain invariants for two-component virtual links.
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