Writhes and $2k$-moves for virtual knots

Let $k$ be a positive integer. A $2k$-move on a knot diagram is a local deformation adding or removing $2k$ half-twists as shown in Figure 1.1. A $2$-move is equivalent to a crossing change; that is, a $2$-move is realized by a crossing change, and vice verse. In this sense, a $2k$-move can be considered as a generalization of a crossing change. The $2k$-moves form an important family of local moves in classical knot theory. In fact, they have been well studied by means of many invariants of classical knots and links; for example, Alexander polynomials [12], Jones, HOMFLYPT and Kauffman polynomials [16], Burnside groups [5, 6] and Milnor invariants [14].

Recently, Jeong, Choi and Kim [8] extended the classical study of $2k$-moves to the setting of virtual knots, which are a generalization of classical knots discovered by Kauffman [9]. Roughly speaking, a virtual knot is an equivalence class of generalized knot diagrams called virtual knot diagrams under seven types of local deformations. We say that two virtual knots $K$ and $K^{\prime}$ are related by a $2k$-move if a diagram of $K^{\prime}$ is a result of a $2k$-move on a diagram of $K$.

For a virtual knot $K$, Kauffman [10] defined an integer-valued invariant $J(K)$ called the odd writhe. Satoh and Taniguchi [17] generalized it to a sequence of integer-valued invariants $J_{n}(K)$ of $K$ called the $n$-writhes. This sequence $\{J_{n}(K)\}_{n\neq 0}$ gives rise to a polynomial invariant $P_{K}(t)$ of $K$ known as the the affine index polynomial [11], which is essentially equivalent to the writhe polynomial [4]. Refer to [3] for a good survey of virtual knot invariants derived from chord index, including $J(K)$, $J_{n}(K)$ and $P_{K}(t)$.

In [8, Theorem 2.3], Jeong, Choi and Kim established a necessary condition for two virtual knots to be related by a finite sequence of $2k$-moves using their affine index polynomials. Examining the proof of [8, Theorem 2.3], we can find another necessary condition for such a pair of virtual knots in terms of the $n$-writhes and odd writhes. The first aim of this paper is to prove the following.

Although the assertion (i) in this theorem is obtained from [8, Theorem 2.3] immediately, the author believes that it is worth stating it as a separate result.

A $\Xi$-move on a virtual knot diagram is a local deformation exchanging the positions of $c_{1}$ and $c_{3}$ of three consecutive real crossings $c_{1}$, $c_{2}$ and $c_{3}$ as shown in Figure 1.2, where we omit the over/under information of each crossing $c_{i}$ $(i=1,2,3)$. The $\Xi$-move arises naturally as a diagrammatic characterization of virtual knots having the same odd writhe. In fact, Satoh and Taniguchi [17] proved the following.

Inspired by this theorem, we use $\Xi$-moves together with $2k$-moves to characterize virtual knots having the same congruence class of odd writhes modulo $2k$. The second aim of this paper is to prove the following.

Although the crossing change is an unknotting operation for classical knots, it was shown by Carter, Kamada and Saito in [1, Proposition 25] that not every virtual knot can be unknotted by crossing changes. This fact justifies the notion of flat virtual knots. A flat virtual knot [9] is an equivalence class of virtual knots up to crossing changes. Equivalently, a flat virtual knot is represented by a virtual knot diagram with all the real crossings replaced by flat crossings, where a flat crossing is a transverse double point with no over/under information.

In [2, Lemma 2.2], Cheng showed that the odd writhe for any virtual knot takes values in even integers. Therefore any virtual knot $K$ and the trivial one $O$ satisfy $J(K)\equiv J(O)\equiv 0\pmod{2}$. By Theorem 1.3 for $k=1$, the two knots $K$ and $O$ are related by a finite sequence of crossing changes and $\Xi$-moves. In other words, we have the following.

The rest of this paper is organized as follows. In Section 2, we review the definitions of a virtual knot, a Gauss diagram, the $n$-writhe and the odd writhe, and prove Theorem 1.1. Section 3 is devoted to the proof of Theorem 1.3. Our main tool is the notion of shell-pairs, which are certain pairs of chords of a Gauss diagram introduced in [13]. In the last section, for two virtual knots $K$ and $K^{\prime}$ that are related by a finite number of $2k$-moves, we study their $2k$-move distance $\mathrm{d}_{2k}(K,K^{\prime})$ defined as the minimal number of such $2k$-moves. We show that for any virtual knot $K$ and any positive integer $a$, there is a virtual knot $K^{\prime}$ such that $\mathrm{d}_{2k}(K,K^{\prime})=a$ (Proposition 4.2).

We begin this section by recalling the definitions of virtual knots and Gauss diagrams from [7, 9].

A virtual knot diagram is the image of an immersion of an oriented circle into the plane whose singularities are only transverse double points. Such double points consist of positive, negative and virtual crossings as shown in Figure 2.1. A positive/negative crossing is also called a real crossing.

Two virtual knot diagrams are said to be equivalent if they are related by a finite sequence of generalized Reidemeister moves I–VII as shown in Figure 2.2. A virtual knot is the equivalence class of a virtual knot diagram. In particular, a classical knot can be considered as a virtual knot diagram with no virtual crossings, called a classical knot diagram, up to the moves I, II and III. In [7, Theorem 1.B], Goussarov, Polyak and Viro proved that two equivalent classical knot diagrams are related by a finite sequence of moves I, II, and III; in other words, the set of virtual knots contains that of classical knots. In this sense, virtual knots are a generalization of classical knots.

A Gauss diagram is an oriented circle equipped with a finite number of signed and oriented chords whose endpoints lie disjointly on the circle. In figures the underlying circle and chords of a Gauss diagram will be drawn with thick and thin lines, respectively. Gauss diagrams provide an alternative way of representing virtual knots. For a virtual knot diagram $D$ with $n$ real crossings (and some or no virtual crossings), the Gauss diagram $G_{D}$ associated with $D$ is constructed as follows. It consists of a circle and $n$ chords connecting the preimage of each real crossing of $D$. Each chord of $G_{D}$ has the sign of the corresponding real crossing of $D$, and it is oriented from the overcrossing to the undercrossing. For a virtual knot $K$, a Gauss diagram of $K$ is defined to be a Gauss diagram associated with a virtual knot diagram of $K$.

A motivation of introducing virtual knot theory comes from the realization of Gauss diagrams. In fact, the construction above defines a surjective map from virtual knot diagrams onto Gauss diagrams, although not every Gauss diagram can be realized by a classical knot diagram. Moreover, this map induces a bijection between the set of virtual knots and that of Gauss diagrams modulo Reidemeister moves I, II and III defined in the Gauss diagram level as shown in Figure 2.3 [7, Theorem 1.A]. Refer also to [9, Section 3.2]. Therefore a virtual knot can be regarded as the equivalence class of a Gauss diagram.

We will use two local deformations on Gauss diagrams as shown in Figure 2.4 as well as the Reidemeister moves I, II and III. These deformations are the counterparts of a $2k$-move and a $\Xi$-move for Gauss diagrams. More precisely, a $2k$-move on a Gauss diagram adds or removes $2k$ chords with the same sign $\varepsilon$ whose initial and terminal endpoints appear alternatively. Let $P_{1}$, $P_{2}$ and $P_{3}$ be three consecutive endpoints of chords of a Gauss diagram. A $\Xi$-move exchanges the positions of $P_{1}$ and $P_{3}$ with preserving the signs $\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}$ and orientations of the chords. In the right of the figure, a pair of dots $\bullet$ marks the two endpoints $P_{1}$ and $P_{3}$ exchanged by a $\Xi$-move.

Using Gauss diagrams, we now define the $n$-writhe and the odd writhe of a virtual knot $K$. For a Gauss diagram $G$ of $K$, let $\gamma$ be a chord of $G$. If $\gamma$ has a sign $\varepsilon$, then we assign $\varepsilon$ and $-\varepsilon$ to the terminal and initial endpoints of $\gamma$, respectively. The endpoints of $\gamma$ divides the underlying circle of $G$ into two arcs. Let $\alpha$ be one of the two oriented arcs that starts at the initial endpoint of $\gamma$. See Figure 2.5. The index of $\gamma$, $\mathrm{ind}(\gamma)$, is the sum of sings of endpoints of chords on $\alpha$. For an integer $n$, we denote by $J_{n}(G)$ the sum of signs of chords with index $n$. In [17, Lemma 2.3], Satoh and Taniguchi proved that $J_{n}(G)$ is an invariant of the virtual knot $K$ for any $n\neq 0$; that is, it is independent of the choice of $G$. This invariant is called the $n$-writhe of $K$ and denoted by $J_{n}(K)$. The odd writhe $J(K)$ of $K$, defined by Kauffman [10], is given by

Refer to [3, 10, 17] for more details.

In [8], Jeong, Choi and Kim prepared Lemma 2.2 to show Theorem 2.3 giving the necessary condition mentioned in Section 1. We rephrase and reprove this lemma from the Gauss diagram point of view for the proof of Theorem 1.1.

Theorem 1.1 together with Theorem 1.2 implies an interesting consequence, which states that the set of $2k$-move equivalence classes of a virtual knot $K$ for all $k\geq 1$ determines the $\Xi$-move equivalence class of $K$.

This section is devoted to the proof of Theorem 1.3. Our main tool is the notion of shell-pairs, which are certain pairs of chords of a Gauss diagram developed in [13] for classifying $2$-component virtual links up to $\Xi$-moves.

Let $P_{1}$, $P_{2}$ and $P_{3}$ be three consecutive endpoints of chords of a Gauss diagram $G$. We say that a chord of $G$ is a shell if it connects $P_{1}$ and $P_{3}$. See the left of Figure 3.1. Note that the orientation of a shell can be reversed by a $\Xi$-move exchanging the positions of $P_{1}$ and $P_{3}$. A positive shell-pair (or negative shell-pair) consists of a pair of positive shells (or negative shells) whose four endpoints are consecutive. See the right of the figure, where we omit the orientations of shells.

We prepare three results (Lemmas 3.1, 3.2 and Proposition 3.3) to give the proof of Theorem 1.3. The first and second results will be used to prove the third one.

The following lemma was shown in [13, 17].

For an integer $a$, let $G(a)$ be the Gauss diagram in Figure 3.5; that is, it consists of $|a|$ shell-pairs with sign $\varepsilon$, where $\varepsilon=1$ for $a>0$ and $\varepsilon=-1$ for $a<0$. In particular, $G(0)$ is the Gauss diagram with no chords. We denote by $K(a)$ the virtual knot represented by $G(a)$. We remark that $J(K(a))=2a$.

We give a normal form of an equivalence class of virtual knots under $2k$-moves and $\Xi$-moves as follows.

Now we are ready to prove Theorem 1.3.

The following result is an immediate consequence of the proof of Theorem 1.3.

This section studies the $2k$-move distance for virtual knots. For two virtual knots $K$ and $K^{\prime}$ that are related by a finite sequence of $2k$-moves, we denote by $\mathrm{d}_{2k}(K,K^{\prime})$ the minimal number of $2k$-moves needed to deform a diagram of $K$ into that of $K^{\prime}$. In particular, we set $\mathrm{u}_{2k}(K)=\mathrm{d}_{2k}(K,O)$, where $O$ is the trivial knot. We will show that for any virtual knot $K$ and any positive integer $a$, there is a virtual knot $K^{\prime}$ such that $\mathrm{d}_{2k}(K,K^{\prime})=a$ (Proposition 4.2).

In [8, Theorem 2.3], Jeong, Choi and Kim gave a lower bound for $\mathrm{d}_{2k}(K,K^{\prime})$ using the affine index polynomials of $K$ and $K^{\prime}$, which can be rephrased in terms of the $n$-writhes as follows.

We remark that this theorem is a direct consequence of Lemma 2.1.

In [8, Example 2.4], Jeong, Choi and Kim demonstrated that the lower bound for $\mathrm{u}_{2k}(K)$ in the theorem is sharp for some virtual knots $K$. However, they did not make it clear whether for a pair of nontrivial virtual knots $K$ and $K^{\prime}$, the lower bound for $\mathrm{d}_{2k}(K,K^{\prime})$ is sharp. We answer this by proving the following.

We remark that for any positive integer $a$, there is a family of infinitely many virtual knots $\{K_{s}\}_{s=1}^{\infty}$ such that $\mathrm{u}_{2k}(K_{s})=a$ $(s\geq 1)$. In fact, let $K_{s}$ be the virtual knot represented by the diagram in [15, Figure 2.6] with $m=ak$. Then by Theorem 4.1, it can be seen that these virtual knots $K_{s}$’s form such a family.