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 [Reference Ishii5], 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 [Reference Ishii6], 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 [Reference Benedetti and Petronio1] 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 [Reference Kauffman10] 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 [Reference Ishii6]. According to a result in [Reference Ishii6], 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.
Local moves.
A notion of quandle is introduced by Joyce [Reference Joyce8], and independently by Matveev [Reference Matveev13]. 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. [Reference Kamada9]). 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 [Reference Ishii6]. 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 [Reference Ishii5] that there exists a 2-disk
$\Sigma $
in M satisfying the following properties as shown in Figure 2:
-
(1)
$|T_{\pm }(\varphi _t,\Sigma )(x)|<\infty $
for any
$x\in M$
. -
(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) 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 [Reference Ishii, Ishikawa, Koda and Naoe7].
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
$3$
.
Flow-spine.
Singularities.
Lemma 2.1 [Reference Ishii5]
$M\setminus P$
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).
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 [Reference Ikeda and Inoue2]. Conversely, we reconstruct M from the
$3$
-ball
$B_P$
by gluing its boundary along the information from the DS-diagram
$(S,\Gamma )$
.
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.
Types of a vertex on an E-cycle e.
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 [Reference Ikeda and Kouno3] 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.
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 [Reference Kauffman10] 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.
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.
Virtual knot diagram from E-data.
We remark that Benedetti and Petronio’s “o-graph” in [Reference Benedetti and Petronio1] is a similar notion to our virtual knot diagram (see also [Reference Koda12]). We also remark that the first author considers an “oriented and coded graph” in [Reference Ishii6]. 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.
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.
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.
$\mathrm {I}_1$
-,
$\mathrm {I}_2$
-, 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.
$\mathrm {I}_0^*$
-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 [Reference Benedetti and Petronio1].
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 [Reference Ishii6], the first author showed the following theorem (cf. [Reference Benedetti and Petronio1, Section 6.3]).
Theorem 4.2 [Reference Ishii6]
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 [Reference Koda12] 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 [Reference Ishii6] 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 [Reference Benedetti and Petronio1] (cf. [Reference Koda12]). 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.
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) For any
$a,b\in X$
, there exists the unique
$c\in X$
such that
$c*b=a$
. -
(2) For any
$a,b,c\in X$
,
$(b*a)*(c*a)=b*c$
. -
(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
$(X,*)$
.
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
$*b$
.
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.
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. [Reference Kawauchi11, Reference Nakamura, Nakanishi, Saito and Satoh14, Reference Nakamura, Nakanishi and Satoh15]). 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.
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.
$L(2,1)$
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)
$L(1,1)$
and
$-L(1,1)$
are not effectively n-risandle colorable for any
$n\geq 0$
. -
(2)
$L(p,1)$
and
$-L(p,1)\ (p\geq 2)$
are effectively p-risandle colorable. -
(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.
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
$L(p',1)$
.
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.
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 [Reference Ishii4] 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.
A E-data for 3-manifolds
In this appendix, we describe E-data of
$3$
-manifolds given in this paper by mainly using their Heegaard diagrams.
A.1 The lens space
$L(p,q)$
Let
$(T;\alpha ,\beta )$
be the standard Heegaard diagram of
$L(p,q)$
, that is,
-
(i) T is a torus decomposing
$L(p,q)$
into two solid tori, say V and W, -
(ii)
$\alpha $
and
$\beta $
are boundaries of meridian disks, say
$D_V$
and
$D_W$
, of V and W, respectively, -
(iii)
$\alpha $
intersects
$\beta $
in p points, and -
(iv)
$[ \alpha ]=[m]$
and
$[ \beta ]=p[ \ell ]+q[m]$
with respect to a standard basis
$[m],[\ell ] \in H_1(T)=\mathbb {Z} \oplus \mathbb {Z}$
.
In this subsection, we obtain a DS-diagram of
$L(p,1)$
with
$p\ge 1$
from the standard Heegaard diagram
$(T;\alpha ,\beta )$
. To this end, we take a perturbed Heegaard diagram
$(T;\alpha ,\beta ')$
at an intersection point of
$\alpha $
and
$\beta $
. We may assume that each component of
$\alpha \cup \beta '$
is successively labeled and oriented such that
$\alpha =\alpha _0 \cup \alpha _1 \cup \cdots \cup \alpha _{p+1}$
and
$\beta '=\beta _0 \cup \beta _1 \cup \cdots \cup \beta _{p+1}$
as shown in Figure 25, which illustrates the case
$p=3$
. We also note that
$\alpha _0 \cup \beta _0$
cuts off a disk, say
$\Delta $
, from the torus T.
A perturbed Heegaard diagram
$(T;\alpha ,\beta ')$
of
$L(3,1)$
.
Let
$V'$
and
$W'$
be solid cylinders obtained by cutting V and W along
$D_V$
and
$D^{\prime }_W$
, respectively, where
$D^{\prime }_W$
is a meridian disk bounded by
$\beta '$
. Then
$V'$
and
$W'$
can be respectively regarded as half-balls with the equator
$\alpha _0\cup \beta _0$
as shown in Figure 26.
A Heegaard diagram to a DS-diagram of
$L(3,1)$
.
Since each boundary of the two half-balls contains a copy of the disk
$\Delta $
bounded by
$\alpha _0 \cup \beta _0$
and the two copies are identified in
$L(p,1)$
, we obtain a
$3$
-ball, say B, by attaching
$V'$
to
$W'$
along those copies of
$\Delta $
. Then
$\Gamma =(\bigcup _i \alpha _i) \cup (\bigcup _j \beta _j)$
in the boundary of the 3-ball B gives a DS-diagram. More precisely, we see that
$L(p,1)$
has a DS-diagram
$(\partial B,\Gamma )$
with the E-cycle
$\alpha _0 \cup \beta _0$
slightly tilted, which is indicated as the red colored curve in Figure 27. We note that all the vertices on the E-cycle are of
$\ell $
-type. Then we obtain the E-data, and hence, its Gauss diagram as shown in the middle and right of Figure 27. We note that one can similarly obtain DS-diagrams of general lens spaces
$L(p,q)$
with
$p\ne 0$
.
A DS-diagram with an E-cycle of
$L(3,1)$
.
A.2 The n-fold cyclic branched cover
$M_n$
of the 3-sphere branched over the trefoil knot
Let
$M_n$
be the n-fold cyclic branched cover of the 3-sphere branched over the trefoil knot. It is known that the left of Figure 28 shows a surgery diagram of
$M_n$
and
$M_3$
is the Quaternion space
$Q^3$
(see [Reference Rolfsen16, Section D of Chapter 10]). Let V be a regular neighborhood of the graph as shown in the middle of Figure 28, which illustrates the case
$n=3$
. Then V is a genus n handlebody. We also see that its exterior, say W, is also a genus n handlebody.
The n-fold cyclic branched cover of the
$3$
-sphere branched over the trefoil knot.
Hence, we can obtain a genus n Heegaard diagram of
$M_n$
with the Heegaard surface
$S=\partial V=\partial W$
and meridian systems
$\mathcal {A}=\bigcup _i \alpha ^i$
,
$\mathcal {B}=\bigcup _i \beta ^i$
as shown in the right of Figure 28, which illustrates the case
$n=3$
. We note that the Heegaard diagram as shown in the left of Figure 29 is obtained from that in the right of Figure 28 by deforming V into the standard form. In this Heegaard diagram, we can find a disk component
$\Delta $
from S cut along
$\mathcal A \cup \mathcal B$
such that each of
$\partial \Delta \cap \alpha ^i$
and
$\partial \Delta \cap \beta ^i$
is a connected arc for all i. Hence,
$(S;\mathcal {A},\mathcal {B};\Delta )$
is a so-called punctured Heegaard diagram considered in [Reference Koda12]. Let V cut along
$\mathcal {A}$
, say
$V'$
, be regarded as a south half-ball. Similarly, let W cut along
$\mathcal {B}$
, say
$W'$
, be regarded as a north half-ball. Then the boundary of the whole
$3$
-ball
$V'\cup _{\Delta } W'$
gives a DS-diagram of
$M_n$
. This implies that we can read the E-data, and hence, its Gauss diagram from this Heegaard diagram as shown in Figure 29 (for details, see [Reference Koda12, Section 2]).
An E-data of the Quaternion space
$Q^3$
.
A.3 The Poincaré homology sphere
The Poincaré homology sphere is constructed from a dodecahedron by identifying its opposite faces as shown in the left of Figure 30 (cf. [Reference Seifert and Threlfall17]). For the Poincaré homology sphere in Proposition 6.5, we take its mirror image on the equator as shown in the middle of Figure 30. As we mentioned in the proof of Corollary 4.3, taking the mirror image changes the orientation of the manifold. This labeled dodecahedron directly gives a DS-diagram with the E-cycle colored red, and hence, the E-data. We note that all the vertices on the E-cycle are of
$\ell $
-type. Thus, we obtain the Gauss diagram in the right of Figure 30.
The Poincaré homology sphere.
B Remarks for I-move and S-move
Originally, the I-move and R2-move for a virtual knot diagram in this paper are introduced in [Reference Ishii6] as “the first regular move and second regular move for an oriented and coded graph,” respectively. An oriented and coded graph is a realization of an E-data of a
$3$
-manifold. The first regular move for an oriented and coded graph is illustrated in the upper of Figure 31. It is presented as a local virtual knot diagram as shown in the lower of Figure 31 by using the manner in this paper.
The first regular move in [Reference Ishii6].
Lemma B.1 The first regular move can be realized by a single I-move and an R2-move.
Proof Figure 32 shows the lemma.
The first regular move and I-move.
The surgery move in [Reference Ishii6] for an oriented and coded graph is illustrated in the upper of Figure 33. It is presented as a local virtual knot diagram as shown in the lower of Figure 33 by using the manner in this paper. Our S-move is also slightly different from that in [Reference Ishii6].
The surgery move in [Reference Ishii6].
Lemma B.2 The surgery move in [Reference Ishii6] can be realized by a single S-move in this paper and a finite sequence of I-moves, R2-moves, virtual Reidemeister moves.
Proof Figure 34 shows the lemma.
Deformation of a surgery move.
