A coloring invariant of 3-manifolds derived from their flow-spines and virtual knot diagrams

Abstract

For a closed, connected, and oriented 3-manifold with a non-singular flow, we construct its virtual knot diagram via a “flow-spine” of the manifold. Then, we introduce a coloring invariant of 3-manifolds through their virtual knot diagrams, and classify some 3-manifolds by using the invariant.

1 Introduction

Throughout this paper, any $3$ -manifold is closed, connected, and oriented unless otherwise stated. It is known that there are several kinds of descriptions of a $3$ -manifold such as a triangulation, a Heegaard diagram, a framed knot or link presentation via Dehn surgery, and so on. Each of them leads us to understand $3$ -manifolds. It is also known that many invariants of a $3$ -manifold are defined and calculated from those descriptions. In this paper, we consider a virtual knot diagram as a description of a $3$ -manifold.

In [5], the first author introduced a notion of a flow-spine of a $3$ -manifold by considering its non-singular flow and described the $3$ -manifold as “E-data” by using a flow-spine. In [6], he also gave local deformations of an E-data and showed that two E-data of orientation-preserving diffeomorphic $3$ -manifolds are related to each other by a finite sequence of the local deformations. Flow-spines are also studied by Benedetti and Petronio in [1] through branched standard spines. In this paper, we construct a virtual knot diagram from an E-data of a $3$ -manifold. In this sense, we say that a $3$ -manifold is presented by a virtual knot diagram. A virtual knot diagram introduced by Kauffman [10] is an oriented circle immersed in the $2$ -plane equipped with only finitely many transversal double points which are either real crossings or virtual crossings.

We also consider local moves of a virtual knot diagram as shown in Figure 1 which correspond to the local deformations of E-data in [6]. According to a result in [6], we see that two virtual knot diagrams constructed from orientation-preserving diffeomorphic $3$ -manifolds are related to each other by a finite sequence of the local moves as shown in Figure 1.

Figure 1: Local moves.

A notion of quandle is introduced by Joyce [8], and independently by Matveev [13]. A quandle is a set with a binary operation which satisfies three axioms corresponding to Reidemeister moves in ordinary knot theory. It is known that a quandle produces a coloring invariant of a knot, which is a generalization of a Fox coloring of a knot (cf. [9]). In this paper, we introduce a notion of risandle inspired by quandle. A risandle $(X,*)$ is a set X with a binary operation $*$ which satisfies three axioms corresponding to the local moves as shown in Figure 1 (see Definition 5.1). We also introduce a coloring invariant of a virtual knot diagram obtained from a risandle. For a $3$ -manifold M and a virtual knot diagram D presenting M, let $\mathrm {c}_X(M;D)$ be the cardinality of a set of colorings by $(X,*)$ .

Theorem 1.1 For a $3$ -manifold M and a risandle $(X,*)$ , $\mathrm {c}_X(M;D)$ does not depend on a particular choice of a virtual knot diagram D presenting M.

By Theorem 1.1, we can denote $\mathrm {c}_X(M)$ instead of $\mathrm {c}_X(M;D)$ .

Theorem 1.2 If two $3$ -manifolds M and $M'$ are orientation-preserving diffeomorphic to each other, then we have $\mathrm {c}_X(M)=\mathrm {c}_X(M')$ .

As applications of Theorem 1.2, our coloring invariants classify lens spaces $L(p,1)\ (p\geq 1)$ and distinguish the Poincaré homology $3$ -sphere from the $3$ -sphere.

This paper is organized as follows: In Section 2, we review a flow-spine and an E-data of a $3$ -manifold. In Section 3, after reviewing a virtual knot diagram and its Gauss diagram, we construct a virtual knot diagram from an E-data of a $3$ -manifold. In Section 4, we introduce three local moves as shown in Figure 1 corresponding to the local deformations of an E-data in [6]. In Section 5, we introduce a notion of risandle and show that a coloring obtained from a risandle is an invariant of a virtual knot diagram under our three local moves. In Section 6, we prove Theorems 1.1 and 1.2 and give several applications. In Appendix A, we describe E-data of $3$ -manifolds appeared in this paper. In Appendix B, we give several remarks with respect to our local moves.

2 E-data of 3-manifold

Let M be a $3$ -manifold, and let $\{\varphi _t\}_{t\in {\Bbb R}}$ be the non-singular flow generated by a non-singular vector field on M. Let $\Sigma $ be a compact $2$ -submanifold of M with boundary such that it is included in some open $2$ -submanifold of M which is nowhere tangential to $\varphi _t$ . We define two functions $T_+(\varphi _t,\Sigma )$ and $T_-(\varphi _t,\Sigma )$ of $x\in M$ such that

$$ \begin{align*} T_+(\varphi_t,\Sigma)(x)&=\inf\{t>0\,|\,\varphi_t(x)\in\Sigma\}\text{ and }\\ T_-(\varphi_t,\Sigma)(x)&=\sup\{t<0\,|\,\varphi_t(x)\in\Sigma\}, \end{align*} $$

where $T_+(\varphi _t,\Sigma )(x)=\infty $ if $\varphi _t(x)\not \in \Sigma $ for any $t>0$ and $T_-(\varphi _t,\Sigma )(x)=-\infty $ if $\varphi _t(x)\not \in \Sigma $ for any $t<0$ . For an $x\in M$ satisfying $T_{+}(\varphi _t,\Sigma )(x)<\infty $ , put $F(x)=\varphi _{\tau }(x)$ , where $\tau =T_+(\varphi _t,\Sigma )(x)$ . It is proved in [5] that there exists a 2-disk $\Sigma $ in M satisfying the following properties as shown in Figure 2:

1. (1) $|T_{\pm }(\varphi _t,\Sigma )(x)|<\infty $ for any $x\in M$ .

2. (2) If $x\in \partial \Sigma $ and $F(x)\in \partial \Sigma $ , then $F(\partial \Sigma )$ intersects transversally with $\partial \Sigma $ at $F(x)$ .

3. (3) If $x\in \partial \Sigma $ and $F(x)\in \partial \Sigma $ , then $F(F(x))\in \mathrm {Int}\Sigma $ .

Such a $2$ -disk $\Sigma $ is called a normal section. See also [7].

Figure 2: Properties (1) and (2).

Let $P=P(\varphi _t,\Sigma )$ be the discontinuity set of $T_+(\varphi _t,\Sigma )(x)$ , that is,

$$\begin{align*}P(\varphi_t,\Sigma)=\Sigma\cup\{\varphi_t(x)\in M\,|\,x\in\Sigma, F(x)\in\partial\Sigma, 0<t\leq T_+(\varphi_t,\Sigma)(x)\}\end{align*}$$

as shown in the upper of Figure 3. We see that P is a compact two-dimensional polyhedron whose point x has a regular neighborhood homeomorphic to one of the three types as shown in Figure 4. We say that a singular point x in the rightmost of Figure 4 is of type $3$ .

Figure 3: Flow-spine.

Figure 4: Singularities.

Lemma 2.1 [5]

$M\setminus P$ is homeomorphic to an open $3$ -ball.

Proof We take two normal sections $\Sigma _1$ and $\Sigma _2$ slightly larger and smaller than $\Sigma $ , respectively, i.e., $\mathrm {Int}\, \Sigma _1 \supset \Sigma $ and $\mathrm {Int}\, \Sigma \supset \Sigma _2$ . Let $\varepsilon $ be a sufficiently small positive real number such that $\{ \varphi _t (x) \,|\, x\in \Sigma _1, \ -\varepsilon < t < \varepsilon \}$ is homeomorphic to $\Sigma _1 \times (-\varepsilon , \varepsilon )$ . We set $V=\{ \varphi _t (x) \,|\, x \in \Sigma _2,\ T_-(\varphi _t,\Sigma _1)(x) +\varepsilon \le t \le -\varepsilon \}$ . Then V is homeomorphic to a subset $\{ (x,t) \,|\, x \in \Sigma _2,\ T_-(\varphi _t,\Sigma _1)(x) +\varepsilon \le t \le -\varepsilon \}$ of $\Sigma _2 \times \mathbb {R}$ . This implies that V is homeomorphic to a 3-ball. Since $\Sigma _1$ and $\Sigma _2$ are slightly larger and smaller than $\Sigma $ , respectively, $M \setminus V$ is a regular neighborhood of $P=P(\varphi _t, \Sigma )$ . Hence, we see that $M\setminus P$ is homeomorphic to an open $3$ -ball (see Figure 5).

Figure 5: Lemma 2.1.

A flow-spine of $(M,\{\varphi _t\})$ is obtained from P by compressing the vertical parts of P and then smoothing the corners as shown in Figure 3. We use the same symbol $P=P(\varphi _t,\Sigma )$ to a flow-spine. Let $B_P$ be the closure of the manifold M cut along P. It follows from Lemma 2.1 that $B_P$ is a $3$ -ball; and, hence, the boundary of $B_P$ , say S, is a $2$ -sphere. We note that a neighborhood of a singular point of type 3 in P is disassembled into four pieces in S as shown in the middle of Figure 6. Then we obtain a graph $\Gamma $ on S constructed from $F(\partial \Sigma )$ and $F^{-1}(\partial \Sigma )$ on $\Sigma $ , and $e=\partial \Sigma $ . The vertices of $\Gamma $ are derived from the singular points of type 3 in P and edges are derived from the images of parts of $\partial \Sigma $ by F. Each edge has an orientation induced by that of $\partial \Sigma $ . We also see that each vertex is trivalent. We consider that e is an oriented cycle of such a graph which lies on the equator of S. Here, we describe $F^{-1}(\partial \Sigma )$ in the north hemisphere and $F(\partial \Sigma )$ in the south hemisphere of S as shown in the right of Figure 6. Such a graph, denoted by $(S,\Gamma )$ , is called a DS-diagram of $(M,\{\varphi _t\})$ with the E-cycle e. The DS-diagram of a $3$ -manifold was originally introduced by Ikeda and Inoue [2]. Conversely, we reconstruct M from the $3$ -ball $B_P$ by gluing its boundary along the information from the DS-diagram $(S,\Gamma )$ .

Figure 6: An $\ell $ -type.

For a DS-diagram of $(M,\{\varphi _t\})$ with an E-cycle e, there are two types of vertices on e as shown in Figure 7, where the upper figure is of $\ell $ -type and the lower is of r-type. A vertex of $\ell $ -type is derived from a singular point of type $3$ as shown in Figure 6, and a vertex of r-type is derived from that as shown in Figure 8.

Figure 7: Types of a vertex on an E-cycle e.

Figure 8: An r-type.

Let $V^{\mathrm {n}}$ be the set of vertices on e incident to an edge in the north hemisphere $(\Sigma ,F^{-1}(\partial \Sigma ))$ , and let $V^{\mathrm {s}}$ be the set of vertices on e incident to an edge in the south hemisphere $(\Sigma ,F(\partial \Sigma ))$ . We remark that $V^{\mathrm {n}}\cup V^{\mathrm {s}}$ is the set of all vertices on e and $V^{\mathrm {n}}\cap V^{\mathrm {s}}=\emptyset $ . Since each singular point of type $3$ of P produces a pair of vertices $v^{\mathrm {n}}\in V^{\mathrm {n}}$ and $v^{\mathrm {s}}\in V^{\mathrm {s}}$ of the same type, there is a one to one correspondence g, called a pairing, from $V^{\mathrm {n}}$ to $V^{\mathrm {s}}$ with $g(v^{\mathrm {n}})=v^{\mathrm {s}}$ . See the right of Figure 6. We consider e as an oriented circle $S^1$ . We obtain a sequence of vertices on e in $V^{\mathrm {n}}\cup V^{\mathrm {s}}$ , each of which is of $\ell $ - or r-type. The sequence up to cyclic permutations equipped with the pairing g is called the E-data of $(M,\{\varphi _t\})$ .

Remark 2.2 It is known [3] that for an abstract E-data, we can construct a compact $3$ -manifold possibly with boundary. Here, an abstract E-data is a sequence of vertices on an oriented circle $S^1$ up to cyclic permutations equipped with information about $\ell $ - or r-type, $v^{\mathrm {n}}$ or $v^{\mathrm {s}}$ , and their pairing.

3 Virtual knot diagram from E-data

A virtual knot diagram is an oriented circle immersed in the $2$ -plane ${\Bbb R}^2$ with only finitely many transversal double points which are either real crossings (positive), (negative), or virtual crossings . The Gauss diagram $G(D)$ of a virtual knot diagram D is an oriented circle with oriented and signed chords, each of which connects the preimage of each double point corresponding to a real crossing of D. Let c be a real crossing of D, and let $\gamma $ be the chord of $G(D)$ corresponding to c. The initial endpoint of $\gamma $ corresponds to the over arc of c, and the terminal endpoint of $\gamma $ corresponds to the under arc of c. The sign of $\gamma $ is the same as that of c. Figure 9 illustrates an example of a virtual knot diagram and its Gauss diagram.

Figure 9: A virtual knot diagram and its Gauss diagram.

In general, there are infinitely many virtual knot diagrams with the same Gauss diagram. However, it is known [10] that they are related to each other by a finite sequence of virtual Reidemeister moves VR1–VR4 as shown in Figure 10 with arbitrary orientations. Thus, we can say that a Gauss diagram determines a virtual knot diagram up to virtual Reidemeister moves.

Figure 10: Virtual Reidemeister moves.

Let M be a $3$ -manifold with a non-singular flow $\{\varphi _t\}$ . The E-data of $(M,\{\varphi _t\})$ can be regarded as an oriented circle $S^1$ with vertices $V^{\mathrm {n}}\cup V^{\mathrm {s}}$ and their pairing g. We can construct the Gauss diagram G from the E-data as follows.

The oriented circle component of G is $S^1$ , and each chord is given by connecting $g(v^{\mathrm {n}})$ and $v^{\mathrm {n}}$ . The chord is oriented from $g(v^{\mathrm {n}})$ to $v^{\mathrm {n}}$ and the sign of the chord is $-1$ if $v^{\mathrm {n}}$ is of $\ell $ -type, $+1$ if $v^{\mathrm {n}}$ is of r-type, respectively. Thus, we obtain a virtual knot diagram D from G, that is, from the E-data of $(M,\{\varphi _t\})$ (see Figure 11). We note that this diagram D is unique up to virtual Reidemeister moves. In this sense, we say that D presents M for a given $3$ -manifold M. We also note that this D depends on a non-singular flow $\{\varphi _t\}$ , that is, a choice of a non-singular vector field on M.

Figure 11: Virtual knot diagram from E-data.

We remark that Benedetti and Petronio’s “o-graph” in [1] is a similar notion to our virtual knot diagram (see also [12]). We also remark that the first author considers an “oriented and coded graph” in [6]. The orientation of a chord of our Gauss diagram is opposite to that of an oriented and coded graph. We adapt it in order to construct a virtual knot diagram from a Gauss diagram according to the manner in ordinary knot theory.

4 Moves for a virtual knot diagram

For a virtual knot diagram, we consider three types of local moves other than virtual Reidemeister moves. The first move is the second Reidemeister move, say R2-move, in ordinary knot theory. We consider four types of R2-moves with respect to the orientation as shown in Figure 12.

Figure 12: R2-moves.

The second move is the following I-move. The $\mathrm {I}_0$ -move is the local move as shown in Figure 13. We consider three kinds of local moves, say $\mathrm {I}_1$ -, $\mathrm {I}_2$ -, and $\mathrm {I}_3$ -move, which are determined by combination of orientations of vertical arcs of the $\mathrm {I}_0$ -move as shown in Figure 13. We also consider $\mathrm {I}_k^*$ -move $(k=0,1,2,3)$ obtained from the $\mathrm {I}_k$ -move by changing over/under information of all the real crossings. We call each of all the patterns the $\mathrm {I}$ -move. We remark that the virtual crossing always appears or disappears in the left-side with respect to the orientation of the horizontal arc.

Figure 13: I-moves.

Lemma 4.1 Any $\mathrm {I}$ -move can be realized by a single $\mathrm {I}_0$ -move and a finite sequence of R2-moves and virtual Reidemeister moves.

Proof First, we show Lemma 4.1 for the $\mathrm {I}_1$ -, $\mathrm {I}_2$ -, and $\mathrm {I}_3$ -moves. The $\mathrm {I}_1$ -move is realized by R2-moves, a second virtual Reidemeister (VR2) move, and a single $\mathrm {I}_0$ -move as shown in the upper of Figure 14. Similarly, the $\mathrm {I}_2$ -move is realized by R2-moves, a VR2-move, and a single $\mathrm {I}_1$ -move, and the $\mathrm {I}_3$ -move is realized by R2-moves, a VR2-move, and a single $\mathrm {I}_0$ -move as shown in the middle and the lower of Figure 14.

Figure 14: $\mathrm {I}_1$ -, $\mathrm {I}_2$ -, and $\mathrm {I}_3$ -moves.

The $\mathrm {I}_0^*$ -move is realized by an R2-move and a VR2-move with a single $\mathrm {I}_2$ -move as shown in Figure 15. Therefore, we see that Lemma 4.1 for the $\mathrm {I}_1^*$ -, $\mathrm {I}_2^*$ -, and $\mathrm {I}_3^*$ -moves can be proved.

Figure 15: $\mathrm {I}_0^*$ -move.

The third move is the surgery move, S-move for short, as shown in Figure 16. This is essentially equivalent to the combinatorial Pontrjagin move by Benedetti and Petronio in [1].

Figure 16: Surgery move.

Two virtual knot diagrams D and $D'$ are said to be RIS-equivalent if D and $D'$ are related to each other by a finite sequence of virtual Reidemeister moves, R2-moves, I-moves, and S-moves. In [6], the first author showed the following theorem (cf. [1, Section 6.3]).

Theorem 4.2 [6]

Let M and $M'$ be closed, connected, and oriented $3$ -manifolds with non-singular flows $\{\varphi _t\}$ and $\{\varphi ^{\prime }_t\}$ , respectively. Let D and $D'$ be virtual knot diagrams obtained from $(M,\{\varphi _t\})$ and $(M',\{\varphi ^{\prime }_t\})$ , respectively. If M is orientation-preserving diffeomorphic to $M'$ , then D and $D'$ are RIS-equivalent.

It follows from Theorem 4.2 that for each M the RIS-equivalence class of D presenting M does not depend on a non-singular flow, that is, a particular choice of a non-singular vector field on M.

Corollary 4.3 If M is diffeomorphic to $M'$ , then either D and $D'$ or $D^{(c)}$ and $D'$ are RIS-equivalent, where $D^{(c)}$ is the virtual knot diagram obtained from D by changing over/under information of all the real crossings.

Proof Let $-M$ be a $3$ -manifold which is orientation-reversing diffeomorphic to M. It is sufficient to show this corollary under the assumption that $-M$ is orientation-preserving diffeomorphic to $M'$ . Then a DS-diagram for $-M$ is obtained from that for M by taking the mirror image on the equator. This operation corresponds to reversing the orientation and changing the signs of all the chords of its Gauss diagram, and hence, changing over/under information of all the real crossings of its virtual knot diagram. Thus, we see that $D^{(c)}$ presents $-M$ . Since we assume that $-M$ is orientation-preserving diffeomorphic to $M'$ , $D^{(c)}$ and $D'$ are RIS-equivalent by Theorem 4.2.

Remark 4.4 Suppose that virtual knot diagrams D and $D'$ are obtained from $(M,\{\varphi _t\})$ and $(M',\{\varphi ^{\prime }_t\})$ , respectively. Koda [12] showed that if D and $D'$ are equivalent up to virtual Reidemeister moves, then M and $M'$ are orientation-preserving diffeomorphic. The first author showed in [6] that M is orientation-preserving diffeomorphic to $M'$ and $\{\varphi _t\}$ is homotopic to $\{\varphi ^{\prime }_t\}$ if and only if D and $D'$ are related to each other by a finite sequence of virtual Reidemeister moves, R2-moves, and I-moves with the “standard” condition in the sense of Benedetti and Petronio [1] (cf. [12]). In that case, we see that the homotopy class of $\{\varphi _t\}$ is changed by an S-move.

5 Risandle

We first remark that if the first Reidemeister move is allowed then any virtual knot diagram can be changed into a virtual knot diagram with no real crossings under RIS-equivalence. We also remark that a first Reidemeister move is realized by third Reidemeister moves in RIS-equivalence (see Figure 17). Since a virtual knot diagram with no real crossings is RIS-equivalent to the trivial virtual knot diagram, any invariant under RIS-equivalence which is also an ordinary virtual knot invariant is constant.

Figure 17: Virtualization of a real crossing.

In this section, we introduce a notion of risandle and a coloring invariant of a virtual knot diagram under RIS-equivalence.

Definition 5.1 Let X be a set with a binary operation $*:X\times X\to X$ . A pair $(X,*)$ is said to be a risandle if it satisfies the following axioms:

1. (1) For any $a,b\in X$ , there exists the unique $c\in X$ such that $c*b=a$ .

2. (2) For any $a,b,c\in X$ , $(b*a)*(c*a)=b*c$ .

3. (3) For any $a\in X$ , $(a*a)*((a*a)*a)=a$ .

We remark that for each $b\in X$ , the map $*b : X\to X$ defined by $x\mapsto x*b$ is a bijection by Axiom (1).

Proposition 5.1 Let G be a group, and define $a*b=ab^{-1}$ for any $a,b\in G$ . Then $(G,*)$ is a risandle.

Proof For (1), we set $c=ab$ for given $a,b\in G$ . Then we have $c*b=cb^{-1}=(ab)b^{-1}=a$ . If there exists $c'$ such that $c'*b=a$ , then $c'=c$ holds by $c'*b=c'b^{-1}=a$ . For (2), we have $(b*a)*(c*a)=(ba^{-1})(ca^{-1})^{-1}=(ba^{-1})(ac^{-1})=bc^{-1}=b*c$ . Let e be the identity element of G. For any $a\in G$ , we have $a*a=e$ . Then we have $(a*a)*((a*a)*a)=e*(e*a)=(a^{-1})^{-1}=a$ for (3).

Let D be a virtual knot diagram. We regard D as a union of arcs whose endpoints are both undercrossing points with ignoring virtual crossings. If D has r real crossings, then D consists of r arcs. Let ${\mathcal A}(D)=\{\alpha _1,\alpha _2,\ldots ,\alpha _r\}$ be the set of arcs of D. For a risandle $(X,*)$ , a map $C:{\mathcal A}(D)\to X$ is called a risandle coloring of D if each crossing of D satisfies the coloring condition $C(\alpha _i)=C(\alpha _k)*C(\alpha _j)$ , where $\alpha _j$ is the over arc, $\alpha _i$ and $\alpha _k$ are the under arcs in the left- and the right-hand side with respect to the orientation of $\alpha _j$ , respectively (see Figure 18). We call an image of each arc by C a color. We denote by $\mathrm {Col}_X(D)$ the set of risandle colorings of D by a risandle $(X,*)$ .

Figure 18: A risandle coloring.

Theorem 5.2 Let $(X,*)$ be a risandle. If two virtual knot diagrams D and $D'$ are RIS-equivalent, then there exists a one-to-one correspondence from $\mathrm {Col}_X(D)$ to $\mathrm {Col}_X(D')$ .

Proof Fix a risandle coloring C of D. We note that any virtual Reidemeister move does not change the color by C of each arc in D into that in $D'$ , and hence, we have the corresponding risandle coloring of $D'$ . Suppose that D, $D'$ are virtual knot diagrams such that $D'$ is obtained from D by a single (i) R2-move, (ii) I-move, or (iii) S-move in a $2$ -disk $U\subset {\Bbb R}^2$ . We note that D and $D'$ are identical to each other on the outside of U.

(i) For an R2-move, the color c in the upper left of Figure 19 satisfies $c*b=a$ , and the color $c'$ in the upper right of Figure 19 satisfies $c'=a*b$ . Thus, we have the unique risandle coloring of $D'$ , which is coincident with that of D on the outside of U by Axiom (1) with the bijectivity of the map $*b$ .

Figure 19: A risandle coloring for an R2-move and an I-move.

(ii) For an I-move, it is sufficient to consider the $\mathrm {I}_0$ -move by Lemma 4.1 and (i) above. We assign the colors a, b, and c to three arcs as shown in the lower of Figure 19. We obtain the colors of the other arcs by the coloring condition. Since $b*c=(b*a)*(c*a)$ holds by Axiom (2), we also have the unique risandle coloring of $D'$ corresponding to that of D.

(iii) For an S-move, it is sufficient to show that the element $b=(a*a)*a$ in Figure 20 satisfies $(b*b)*b=a$ . By Axiom (2), we have

$$\begin{align*}b*b=((a*a)*a)*((a*a)*a)=(a*a)*(a*a)=a*a.\end{align*}$$

Then we see that $(b*b)*b=(a*a)*((a*a)*a)=a$ by Axiom (3). Therefore, we have the unique risandle coloring of $D'$ corresponding to that of D.

Figure 20: A risandle coloring for an S-move.

For an integer $n\geq 0$ and the additive group ${\Bbb Z}/n{\Bbb Z}$ , we obtain the risandle $({\Bbb Z}/n{\Bbb Z}, *)$ , where $a*b=a-b\in {\Bbb Z}/n{\Bbb Z}$ for $a,b\in {\Bbb Z}/n{\Bbb Z}$ by Proposition 5.1. We call it the n-risandle. A risandle coloring of a virtual knot diagram D by $({\Bbb Z}/n{\Bbb Z}, *)$ is called an n-risandle coloring. An n-risandle coloring is said to be trivial if all the arcs of D are colored by $0\in {\Bbb Z}/n{\Bbb Z}$ , otherwise nontrivial. A virtual knot diagram D is said to be n-risandle colorable if D admits a nontrivial n-risandle coloring.

Since $({\Bbb Z}/n{\Bbb Z}, *)=({\Bbb Z}, *)$ for $n=0$ , we call it a ${\Bbb Z}$ -risandle coloring. We remark that the $1$ -risandle coloring is the trivial coloring. An n-risandle coloring $C\ (n\geq 2)$ of D is effective if the composition $\pi _p\circ C$ is a nontrivial p-risandle coloring of D for any prime factor p of n (cf. [11, 14, 15]). Here, $\pi _p$ is a natural projection ${\Bbb Z}/n{\Bbb Z}\to {\Bbb Z}/p{\Bbb Z}$ . A virtual knot diagram D is said to be effectively n-risandle colorable if D admits an effective n-risandle coloring. If n is prime, then the effective n-risandle colorability is coincident with the n-risandle colorability.

Example 5.3 The left-handed trefoil knot diagram as shown in the left of Figure 21 is effectively $2$ -risandle colorable. The diagram as shown in the right of Figure 21 is effectively $4$ -risandle colorable.

Figure 21: Examples of n-risandle colorings.

It follows from the proof of Theorem 5.2 that if a virtual knot diagram D is effectively n-risandle colorable, then a virtual knot diagram obtained from D under RIS-equivalence is also effectively n-risandle colorable.

Corollary 5.4 The effective n-risandle colorability of a virtual knot diagram is invariant under RIS-equivalence.

6 Risandle colorings of 3-manifolds

Let $(X,*)$ be a risandle. For a $3$ -manifold M and a virtual knot diagram D presenting M, let $\mathrm {c}_X(M;D)$ be the cardinality of $\mathrm {Col}_X(D)$ .

Proof of Theorem 1.1 Let $D_1$ and $D_2$ be virtual knot diagrams presenting M. Since $D_1$ and $D_2$ are RIS-equivalent by Theorem 4.2, there is a one-to-one correspondence from $\mathrm {Col}_X(D_1)$ to $\mathrm {Col}_X(D_2)$ by Theorem 5.2. Then we have $\mathrm {c}_X(M;D_1)=\mathrm {c}_X(M;D_2)$ .

Let $\mathrm {c}_X(M)$ denote $\mathrm {c}_X(M;D)$ for a $3$ -manifold M and a virtual knot diagram D presenting M. Then Theorem 1.2 also follows from Theorems 4.2 and 5.2.

Example 6.1 The lens space $L(2,1)$ is presented by the virtual knot diagram D as shown in the upper of Figure 22 (see Appendix A.1), and has only two $2$ -risandle colorings, that is, $\mathrm {c}_{\mathbb {Z}/2\mathbb {Z}}(L(2,1))=2$ . We also see that $\mathrm {c}_{\mathbb {Z}/2\mathbb {Z}}(-L(2,1))=2$ by considering $2$ -risandle colorings of $D^{(c)}$ . Let $Q^3$ be the Quaternion space, that is, a $3$ -manifold whose fundamental group is the quaternion group. The Quaternion space $Q^3$ is presented by the left-hand trefoil knot diagram as shown in the lower of Figure 22 (see Appendix A.2), and has four $2$ -risandle colorings. Since $\mathrm {c}_{\mathbb {Z}/2\mathbb {Z}}(Q^3)=4\neq \mathrm {c}_{\mathbb {Z}/2\mathbb {Z}}(\pm L(2,1))$ , we see that $L(2,1)$ is not diffeomorphic to $Q^3$ by Theorem 1.2 and Corollary 4.3.

Figure 22: $L(2,1)$ and Quaternion space.

By Theorems 4.2 and 5.2 and Corollary 5.4, we have the following.

Lemma 6.2 The effective n-risandle colorability is invariant of M up to orientation-preserving diffeomorphisms.

A $3$ -manifold M is said to be effectively n-risandle colorable if a virtual knot diagram D presenting M is effectively n-risandle colorable.

Proposition 6.3 For a lens space $L(p,1)\ (p\geq 1)$ , we have the following:

1. (1) $L(1,1)$ and $-L(1,1)$ are not effectively n-risandle colorable for any $n\geq 0$ .

2. (2) $L(p,1)$ and $-L(p,1)\ (p\geq 2)$ are effectively p-risandle colorable.

3. (3) If $L(p,1)$ and $-L(p,1)\ (p\geq 2)$ are effectively n-risandle colorable, then p is divisible by n.

Proof A lens space $L(p,1)$ is presented by a virtual knot diagram $D_p$ as shown in the upper of Figure 23 (see Appendix A.1) and $-L(p,1)$ is presented by a virtual knot diagram $D_p^{(c)}$ as shown in the lower of Figure 23.

Figure 23: Virtual knot diagrams $D_p$ and $D_p^{(c)}$ .

(1) The virtual knot diagram $D_1$ consists of one arc with one real crossing. So we see that $D_1$ admits only the trivial n-risandle coloring for any n. This also holds for $D_1^{(c)}$ .

(2) If $p\geq 2$ , we have p-risandle colorings of $D_p$ and $D^{(c)}_p$ as shown in the upper left and the lower left of Figure 23, respectively. We see that these p-risandle colorings have distinct p colors, respectively, therefore they are effective.

(3) Let C be an effective n-risandle coloring of $D_p$ such that the over arc has the color $x\in {\Bbb Z}/n{\Bbb Z}$ as shown in the upper right of Figure 23. Then we see that $px\equiv 0\pmod n$ holds. Since $(x,n)=1$ holds by the effectivity of C, p is divisible by n. This also holds for $D_p^{(c)}$ as shown in the lower right of Figure 23.

Corollary 6.4 If $p\neq p' \ (\geq 1)$ , then $L(p,1)$ is not diffeomorphic to $L(p',1)$ .

Proof This follows from Corollary 4.3, Lemma 6.2, and Proposition 6.3, immediately.

The Poincaré homology $3$ -sphere PHS, which has the same homology groups as those of the $3$ -sphere $S^3$ , is presented by the virtual knot diagram as shown in the left of Figure 24 (see Appendix A.3). We note that this is not n-risandle colorable for any n. Let $\mathrm {SL}(2,{\Bbb Z}/5{\Bbb Z})$ be the special linear group of degree $2$ over ${\Bbb Z}/5{\Bbb Z}$ . We see that $(\mathrm {SL}(2,{\Bbb Z}/5{\Bbb Z}),*)$ with $A*B=AB^{-1}$ for $A,B\in \mathrm {SL}(2,{\Bbb Z}/5{\Bbb Z})$ is a risandle by Proposition 5.1.

Figure 24: Poincaré homology $3$ -sphere and $S^3$ .

Proposition 6.5 For a risandle $(\mathrm {SL}(2,{\Bbb Z}/5{\Bbb Z}),*)$ defined as above, the Poincaré homology $3$ -sphere PHS has $\mathrm {c}_{\mathrm {SL}(2,{\Bbb Z}/5{\Bbb Z})}(\textit {PHS})>1$ .

Proof As shown in the left of Figure 24, let $A,B,C,D,E\in \mathrm {SL}(2,{\Bbb Z}/5{\Bbb Z})$ be colors of the five arcs, respectively. They have to satisfy coloring conditions $AB^{-1}=E$ , $BC^{-1}=A$ , $CD^{-1}=B$ , $DE^{-1}=C$ , and $EA^{-1}=D$ at each real crossing. If all of $A,B,C,D,E$ are the unit matrix $I=\left ( \begin {array}{cc} 1 & 0\\ 0 & 1 \end {array} \right )$ , then we have a risandle coloring. Furthermore, we see that elements $A=\left ( \begin {array}{cc} 0 & 1\\ 4 & 3 \end {array} \right )$ , $B=\left ( \begin {array}{cc} 0 & 4\\ 1 & 3 \end {array} \right )$ , $C=\left ( \begin {array}{cc} 4 & 4\\ 0 & 4 \end {array} \right )$ , $D=\left ( \begin {array}{cc} 2 & 1\\ 1 & 1 \end {array} \right )$ , and $E=\left ( \begin {array}{cc} 4 & 0\\ 4 & 4 \end {array} \right )$ also satisfy the conditions. Thus, we have $\mathrm {c}_{\mathrm {SL}(2,{\Bbb Z}/5{\Bbb Z})}(\textit {PHS})>1$ .

Remark 6.6 Ishii [4] tells us that $\mathrm {c}_{\mathrm {SL}(2,{\Bbb Z}/5{\Bbb Z})}(PHS)=121$ by computer experiments. On the other hand, the $3$ -sphere $S^3$ is presented by the virtual knot diagram as shown in the right of Figure 24. By the coloring condition, the color $Y\in \mathrm {SL}(2,{\Bbb Z}/5{\Bbb Z})$ satisfies $Y=YY^{-1}=I$ , and hence $\mathrm {c}_{\mathrm {SL}(2,{\Bbb Z}/5{\Bbb Z})}(S^3)=1$ . We also see that $\mathrm {c}_{\mathrm {SL}(2,{\Bbb Z}/5{\Bbb Z})}(-S^3)=1$ . Therefore, it follows from Theorem 1.2 and Corollary 4.3 that the Poincaré homology $3$ -sphere is not diffeomorphic to the $3$ -sphere $S^3$ .

Remark 6.7 It follows from Remark 4.4 that an invariant of a virtual knot diagram under R2-moves and I-moves with the “standard” condition is an invariant of a pair of a $3$ -manifold and a homotopy class of a non-singular flow. By the proof of Theorem 5.2, we see that Axioms (1)–(3) in Definition 5.1 correspond to an R2-move, an I-move, and an S-move, respectively. Therefore, if there exists a set with a binary operation satisfying Axioms (1) and (2) but not Axiom (3), then it may produce an invariant which can distinguish homotopy classes of non-singular flows of a given $3$ -manifold.