数理解析研究所講究録
第 1866 巻 2013 年 105-119

105

A table of coherent band-Gordian distances between knots
Taizo Kanenobu (Osaka City University)
Hiromasa Moriuchi (OCAMI)
Abstract

We introduce some criteria for two links, which are related by a coherent band surgery,using the determinant, and the Jones, HOMFLYPT, and polynomials. We give a table ofcoherent band-Gordian distances between two knots with up to seven crossings.
$Q$

1

Introduction

There are several criterion for two links, which are related by a band surgery or crossingchange. In this paper, we introduce further criteria using the determinant, and the Jones,
HOMFLYPT, and polynomials. band surgery and a crossing change are local changes
in a link diagram as shown in Figure. 1. If we consider oriented links, there are two types
for a band surgery according to an orientation; a coherent band surgery (Fig 2) and an
incoherent one. In particular, an incoherent band surgery between two knots is called an$H(2)$ -move [14] (Figure. 3). Recently, these local moves are studied in connection with
an application to the study of DNA site-specific recombination; see [5, 6, 9].
$A$

$Q$

$)$

$(*\phi\wedge\vee$Figure 1:$111111)\backslash \prime--\backslash --,$$(\backslash 1\backslash _{--}\prime^{--}/11111$$A$

band surgery and a crossing change.

$\infty$

Figure 2:$\infty 111111’\backslash --\backslash --\prime\underline{\mapsto}_{\backslash _{-arrow}}^{\backslash }1\prime^{--}\prime 11111$$A$

coherent band surgery.

Given two links and , we want to decide whether they are related by a band
surgery or a crossing change. The signature and Arf invariant are most useful tools forthis problem (Propositions 2.2 and 2.3). There are also several other methods to deal
$L$

$L’$

106
with this problem: for a coherent band surgery, see [19, 21]; for a crossing change, see[30, 32, 35, 40, 41, 42]; for an $H(2)$ -move, see [20, 23, 26]; see also [1].

$\sim$$\sim$$1111_{1}11 \backslash \backslash \prime\prime\backslash \prime\backslash \backslash \backslash \backslash \backslash \bigcup_{/ ,\backslash \backslash \prime\prime\prime}/\prime\backslash \backslash \backslash \prime’\prime\prime\prime II|$$111||I)_{;\backslash \prime\prime\prime}(1|1\backslash \backslash \prime\prime\backslash \backslash \backslash ;_{\backslash ,\backslash _{-\prime}}\prime\prime\backslash ,\prime\prime./\backslash 111$

Figure 3: An $H(2)$ -move.

Our main results are two criteria: The first one is a condition on the determinant of alink or knot which is obtained from a 2-bridge knot by a coherent band surgery or $H(2)-$move (Theorem 3.2), which is easily obtained by using a condition on the determinant ofa knot obtained from a 2-bridge knot by a crossing change due to Hitoshi Murakami [32](Proposition 3.1).
The second one uses some special values of the polynomial invariants. For the Jones
polynomial, we have a criterion on two links which are related by a coherent band surgery[19, Theorem 2.2] (Theorem 4.2). Developing this, we obtain Theorem 4.6. In a similarway, for the HOMFLYPT polynomial we obtain Theorem 5.4 developing Proposition 5.1,
and for the polynomial Theorem 6.2 developing Proposition 6.1. We give some examples
for each of these criteria, which display the efficiency of them. In a forthcoming paper[24] we will make a detailed report on these criteria.
Notation. For knots and links with up to 9 crossings we use Rolfsen notations [38,Appendix ]. For a knot or link , we denote by $L!$ its mirror image. For an oriented 2component link with crossings we use the notations
and , where we usually supposethat linking number of is negative and that ofis positive as in Table 2 in [21]; more
precisely,
denotes an oriented link with negative linking number with diagram as in
the table of [38] anddenotes one of the oriented links obtained fromby reversing
the orientation of one component.
$Q$

$C$

$L$

$c$

$c_{n}^{2}$$c_{n}^{2’}$$c_{n}^{2’}$$c_{n}^{2}$$c_{n}^{2}$$c_{n}^{2’}$2$c_{n}^{2}$

Some invariants

The Conway polynomial $\nabla(L;z)\in Z[z][4]$ , the Jones polynomial $V(L;t)\in Z[t^{\pm 1/2}][17],$and the HOMFL $YPT$ polynomial $P(L;v, z)\in Z[v^{\pm 1}, z^{\pm 1}][10,17,36]$ are invariants of theisotopy type of an oriented link , which are defined by the following formulas:$L$$\nabla(U;z)=1$ ;$\nabla(L_{+};z)-\nabla(L_{-};z)=z\nabla(L_{0};z)$(1);$V(U;t)=1$ ;$t^{-1}V(L_{+};t)-tV(L_{-};t)=(t^{1/2}-t^{-1/2})V(L_{0};t)$ ;

(2)
(3)

$P(U;v, z)=1$ ;

(4)
(5)

$v^{-1}P(L_{+};v, z)-vP(L_{-};v, z)=zP(L_{0};v, z)$ ,

(6)

107

$L+ L_{-} L_{0}$Figure 4:$A$skein triple.

where is the unknot andis a skein triple.
For a skein triple, the link
is obtained fromby changing a crossing,and vice versa, and the link
is obtained from $L+$ orby a coherent band surgery,and vice versa. Conversely, it is easy to see the following:
Lemma 2.1. If a -component link and $a(c+1)$ -component link $M$ are related by acoherent band surgery, then there exist -component links
and $(c+1)$ -component
links $M+,$ $M$-such that each of the following is a skein triple: $(L_{+}, L, M),$ $(L, L_{-}, M)$ ,$(M_{+}, M, L),$ $(M, M_{-}, L)$ .
For a -component link $L,$ $i^{c-1}V(L;-1)$ is an integer and the determinant $\det L$ isgiven by $\det L=|V(L;-1)|$ . Putting $t=-1$ in Eq. (4), we obtain$U$$(L_{+}, L_{-}, L_{0})$$(L_{+}, L_{-}, L_{0})$$L_{-}$$L_{+}$$L_{-}$$L_{0}$

$L$

$c$

$c$

$L_{+},$$L_{-}$

$c$

-

Let

$(L_{+}, L_{-}, L_{0})$$V(L_{+};-1)+V(L_{-};-1)=2iV(L_{0};-1)$ ;(7)

be a skein triple. Then Murasugi [34, Lemma 7.1] has shown:$|\sigma(L_{\pm})-\sigma(L_{0})|\leq 1$

Since we may consider the link

$L_{+}$or$L_{-}$and vice versa, we have the following.

as obtained from

Proposition 2.2. (i) If two oriented links
then

$L$

(ii) If two oriented links

and

$L’$

(8)
$L_{0}$

by a coherent band surgery,and $L’$ are related by a coherent band surgery,$|\sigma(L)-\sigma(L’)|\leq 1$
$L$

.

.

(9)

are related by a crossing change, then$|\sigma(L)-\sigma(L’)|\leq 2$

.

(10)

The $Arf$ invariant (or Robertello invariant) [37] of a knot $K$ , Arf $(K)$ , is given by
Arf $(K)=a_{2}(K)\in Z_{2}$ ,(11)where $a_{2}(K)$ is the coefficient of of the Conway polynomial of $K$ . Whenever an equality
in this paper contains an Arf invariant it is to be understood in the sense of $mod 2$ . We
say that an oriented link is related (in the sense of Robertello [37]) to a knot $K$ if there
exists a smooth embedding of a planar surface insuch that meets $S^{3}\cross\{0,1\}$transversely in $K$ and , respectively. Let
be a proper link, that is, the sum of thelinking numbers of any component of with all the other components is even. We may
define its Arf invariant to be the Arf invariant of any knot $K$ related to it. In particular,
we have:
Proposition 2.3. If a knot $K$ is obtained from a proper 2-component link by a coherent
band surgery, then Arf $(K)=$ Arf $(L)$ .
$z^{2}$

$L$

$F$

$L$

$S^{3}\cross I$

$F$

$L$

$L$

$L$

108
3

Determinant of a link obtained from a 2-bridge knot by aband surgery

denote the 2with $p>q>0$ and odd, we let
For relatively prime integers$(p,q)$
of . More
space
of
is
the
2-foldbranched
cover
type
bridge knot for which the lens
explicitly, letbe positive integers obtained from the continued fraction$p,$$S_{p,q}$

$p$

$q$

$S^{3}$$a_{1},$$a_{2},$$a_{3},$$\ldots,$$a_{n}$$\frac{p}{q}=a_{1}+\frac{1}{1}$ .

(12)$a_{2}+\overline{a_{3}+\frac{1}{+\frac{1}{a_{n}}}}$

Then
is isotopic to a 2-bridge knot in Conway’s normal formas shown in Figure. 5, where the box containing an integer or $-a,$ $a>0$ , denotes a; cf. [25,presents the mirror image of
2-braid as shown in Figure. 6. Also,Sec. 2.1].
$C(a_{1}, a_{2}, a_{3}, \ldots, a_{n-1}, a_{n})$$S_{p,q}$$a$$S_{p,-q}$$S_{p,q}$Figure 5: The 2-bridge knot $C(a_{1}, a_{2}, a_{3}, \ldots, a_{n-1}, a_{n})$ .

$a$crossings$a$crossings

Figure 6: 2-braids.

The following criteria is due to H. Murakami [32, Corollary 2.8].
Proposition 3.1. Suppose that a knot $K$ is obtained from a 2-bridge knot
crossing change. Then there exists an integer such that:$S_{p,q}$

by a

$s$

$|\det K-p|/2\equiv\pm qs^{2} (mod p)$ .

(13)

Using this, we may deduce the following.

Theorem 3.2. Suppose that a link is obtained from a 2-bridge knot
or incoherent band surgery. Then there exists an integer such that:$L$$S_{p,q}$by a coherent$s$$\det L\equiv\pm qs^{2} (mod p)$

.

(14)

109
Proof. Suppose that

there exists a knot

$L$

$K$

and
are related by a coherent band surgery. Then by Lemma 2.1
such that $(K, S_{p},{}_{q}L)$ is a skein triple. From Eq. (7) we have
$S_{p,q}$$V(K;-1)+V(S_{p,q};-1)=2iV(L;-1)$ ,(15)$2 \det L=|2iV(L;-1)|=|-V(K;-1)+V(S_{p,q};-1)|$ .

(16)

-

which implies

Since

$K$

integer$s$

andare related by a crossing change, by Proposition 3.1 there exists ansuch that Eq. (13) holds, which implies$S_{p,q}$$\det K+p\equiv\det K-p\equiv\pm 2qs^{2} (mod 2p)$ ,(17)

Since $\det K=|V(K;-1)|$ and $p=|V(S_{p,q};-1)|$ , combining Eqs. (16) and (17), we obtain
Eq. (14).
$\square$

By Theorem 3.2 a 2-bridge knot may have some condition on the values of $\det L$ , whereis either a 2-component link with $d_{cb}(S_{p},{}_{q}L)=1$ or a knot with $d_{2}(S_{p},{}_{q}L)=1$ . For
2-bridge knots with up to 8 crossings, Table 1 lists these values; the remaining 2-bridge
knots $3_{1},5_{2},6_{2},7_{1},7_{2},7_{6},8_{4},8_{6},8_{7},8_{14}$ have no such restrictions.
$L$

Table 1: Values which $\det L$ does not take with $d_{cb}(S_{p},{}_{q}L)=1$ or $d_{2}(S_{p)}{}_{q}L)=1$

Example 3.3. Table 2 shows 2-component links which are not obtained from the 2-bridge
knots in Table 1 by a coherent band surgery. The symbol means that the link in therow is not obtained from the 2-bridge knot in the column by a coherent band surgery. For
example, the knot
and the link are not related by a coherent band surgery; moreover
this implies that $K\in\{6_{1},6_{1}!\}$ andare not related by a coherent
band surgery; cf. [15, Table 2], [16, Table II].
$\cross$$6_{1}$$6_{1}^{2}$$L\in\{6_{1}^{2},6_{1}^{2’}, 6_{1}^{2}!, 6_{1}^{2’}!\}$

110
Table 2: Links and 2-bridge knots which are not related by a single coherent band surgery.

4

Coherent band-Gordian distance

The following is Proposition 2.3 in [22]:Proposition 4.1. If two knots $K$ and $K’$ are related by a sequence of two coherent
band surgeries, then they are related by a single $SH(3)$ -move, and vice versa. Thus$d_{cb}(K, K’)=2sd_{3}(K, K’)$ and $u_{cb}(K)=2su_{3}(K)$ .
The following is Theorem 2.2 in [19].
Theorem 4.2. If two links
then

$L$

and

$L’$

are related by a coherent band surgery, $d_{cb}(L, L’)=1,$$V(L;\omega)/V(L’;\omega)\in\{\pm i, -\sqrt{3}^{\pm 1}\}$

.

(18)

Then we have the following, which is given in [22, Theorem 3.1].
Corollary 4.3. If two knots $K$ and $K’$ are related by a single $SH(3)$ -move, $sd_{3}(K, K’)=$, then$V(K;\omega)/V(K’;\omega)\in\{\pm 1, \pm i\sqrt{3}^{\pm 1},3^{\pm 1}\}$ .
(19)
$1$

111

$\ovalbox{\tt\small REJECT}$’$\circ_{\backslash }^{I}\prime\backslash \backslash \backslash \backslash C^{l},\backslash ---\prime/$$SH(3)$-move$\ovalbox{\tt\small REJECT} ||$$\infty$

Figure 7: An $SH(3)$ -move is correspond to two coherent band surgeries.

Example 4.4. Let $K=4_{1}$ and $K’=3_{1}!\# 3_{1}$ . Then $sd_{3}(K, K’)>1$ ; see [15, Table 1].
Since $\sigma(K)=\sigma(K’)=0$ , the signature cannot show $sd_{3}(K, K’)>1$ . However, since$V(K;\omega)=-1,$ $V(K’;\omega)=3$ , we can prove by using Corollary 4.3. In Table 3 we list all
such pairs of knots with up to 7 crossings.
Table 3: Pairs of knots $K$ and $K’$ with $|\sigma(K)-\sigma(K’)|\leq 2$ and $sd_{3}(K, K’)>1.$$\overline{\frac{KK’\sigma(K)\sigma(K’)V(K;\omega)V(K’;\omega)}{4_{1}3_{1}!\# 3_{1}00-13}}$$5_{2} 3_{1}!\# 3_{1} 2 0 -1 3$$7_{6} 3_{1}!\# 3_{1} 2 0 -1 3$$6_{2} 3_{1}\# 3_{1} 2 4 1 -3$$7_{2} 3_{1}\# 3_{1} 2 4 1 -3$$7_{3}! 3_{1}\# 3_{1} 4 4 1 -3$
The following is Theorem 5.2 in [24].

Theorem 4.5. Suppose that $a(c+1)$ -component link is obtained from a -component
link by a coherent band surgery. If $V(L’;\omega)=\eta iV(L;\omega)=\pm i^{c}(i\sqrt{3})^{\delta},$ $\eta=\pm 1$ , then
$L’$

$c$

$L$

$i^{c}V(L’;-1)\equiv\eta i^{c-1}V(L;-1)(mod 3^{\delta+1})$.

Theorem 4.6. Suppose that two links and $L’$ are related by a sequence of two coherent
band surgeries, $d_{cb}(L, L’)=2$ . Let be a -component link. If $V(L;\omega)=-V(L’;\omega)=$, then
$L$

$L$

$c$

$\pm i^{c-1}(i\sqrt{3})^{\delta}$$i^{c-1}V(L;-1)\equiv-i^{c-1}V(L’;-1) (mod 3^{\delta+1})$

(20)

112
By Proposition 4.1, we have:
Corollary 4.7. If two knots $K$ and $K’$ are related by a single $SH(3)$ -move, $sd_{3}(K, K’)=$, and $V(K;\omega)=-V(K’;\omega)=\pm(i\sqrt{3})^{\delta}$ , then$1$

$V(K;-1)\equiv-V(K’;-1) (mod 3^{\delta+1})$(21)

Example 4.8. Let $K=6_{1}$ and $K’=3_{1}$ . Then $sd_{3}(K, K’)>1$ . Since $\sigma(K)=0,$$V(K;\omega)=i\sqrt{3},$$\sigma(K’)=2$ , the signature cannot show $sd_{3}(K, K’)>1$ . However, since$V(K’;\omega)=-i\sqrt{3},$ $V(K;-1)=9,$ $V(K’;-1)=-3$ , we can prove by using Corollary 4.7.
In Table 4 we list all such pairs of knots with up to 7 crossings.
Table 4: Pairs of knots $K$ and $K’$ with $|\sigma(K)-\sigma(K’)|\leq 2$ and $sd_{3}(K, K’)>1.$$\overline{\frac{KK’\sigma(K)\sigma(K’)V(K;\omega)V(K’;\omega)V(K;-1)V(K’;-1)}{6_{1}3_{1}02i\sqrt{3}-i\sqrt{3}9-3}}$$6_{1} 7_{4} 0 -2 i\sqrt{3} -i\sqrt{3} 9 -15$
$6_{1} 7_{7} 0 0 i\sqrt{3} -i\sqrt{3} 9 21$$6_{1} 3_{1}!\neq 4_{1} 0 -2 i\sqrt{3} -i\sqrt{3} 9 -15$

$7_{4}! 7_{7} 2 0 i\sqrt{3} -i\sqrt{3} -15 21$$7_{7}! 7_{7} 0 0 i\sqrt{3} -i\sqrt{3} 21 21$$\underline{3_{1}\neq 4_{1}7_{7}20i\sqrt{3}-i\sqrt{3}-1521}$
Similarly, we have:

and $L’$ are related by a sequence of two
Corollary 4.9. If two 2-component links
coherent band surgeries, $d_{cb}(L, L’)=2$ , and $V(L;\omega)=-V(L’;\omega)=\pm i(i\sqrt{3})^{\delta}$ , then$L$

$V(L;-1)/i\equiv-V(L’;-1)/i (mod 3^{\delta+1})$(22)

In Table 4 we list all pairs of 2-component links with up to 6 crossings, which canbe shown to have coherent band-Gordian distance $>2$ by Corollary 4.9 but cannot beshown by using the signature. Thus by Table 3 in [15] we can conclude they have coherent
band-Gordian distance 4.
Table 5: Pairs of links$L$and $L’$ with $|\sigma(L)-\sigma(L’)|\leq 2$ and $d_{cb}(L, L’)=4.$$\overline{\frac{LL’\sigma(L)\sigma(L’)V(L;\omega)V(L’;\omega)V(L;-1)/iV(L’;-1)/i}{3_{1}\neq H_{+}6_{3}^{2}13-\sqrt{3}\sqrt{3}6-12}}$$3_{1}\# H+ 6_{3}^{2’} 1 -1 -\sqrt{3} \sqrt{3} 6 -12$
$T_{6’}! 6_{3}^{2} 1 3 -\sqrt{3} \sqrt{3} 6 -12$$\underline{T_{6^{J}}!6_{3}^{2’}1-1-\sqrt{3}\sqrt{3}6-12}$

113
5

The HOMFLYPT polynomialLet
be the -fold cyclic covering space ofMillett [27, Theorem 2] have shown:$k$$\Sigma_{k}(L)$$S^{3}$

branched over a link$L$

. Lickorish and$P(L;i, i)=(-2)^{\tau/2}$ ,(23)

where $\tau=\dim H_{1}(\Sigma_{3}(L);Z_{2})$ . Putting $v=z=i$ in Eq. (6), we obtain$P(L_{+};i, i)+P(L_{-};i, i)+P(L_{0};i, i)=0$ ,

(24)

where
is a skein triple. Using this, we have a criterion on the HOMFLYPT
polynomials of two links which are related by a crossing change [29, Theorem 1.1] or acoherent band surgery [21, Proposition 2.4].
$(L_{+}, L_{-}, L_{0})$Proposition 5.1. If two links
coherent band surgery, then

$L$

and

$L’$

are related by either a crossing change or a$P(L;i, i)/P(L’;i, i)\in\{1, -2^{\pm 1}\}$ .

(25)

The Conway polynomial $\nabla(L;z)$ of a -component link may be written $\nabla(L;z)=$is an integer polynomial in . Then we obtain a symmetric integer
polynomial
by
,
(26)
which is called the Hosokawa polynomial [12]; cf. [33, pp. 120]. Then Hosokawa andKinoshita [13] have shown the following; cf. [28, Corollary 9.8]:
$L$

$c$

$z^{c-1}\varphi(z),$ $wh_{\sim}ere\varphi(z)$$z^{2}$$\triangle_{L}(t)$$\triangle_{L}(t)=\varphi(t^{1/2}-t^{-1/2})\sim$

Proposition 5.2. The order of the first homology group of the -fold cyclic covering spaceof branched over a -component link $L,$ $H_{1}(\Sigma_{k}(L);Z)$ , is given by$k$$S^{3}$$c$$k^{c-1} \prod_{j=1}^{k-1}\triangle_{L}(\xi^{j})\sim$where$\xi w$,(27)

a primitive $kth$ root of unity.

Using Proposition 5.2, we obtain:

Lemma 5.3. Let$L$

be a -component link. If $P(L;i, i)=(-2)^{h}$ , then$c$$[\nabla(L;z)/z^{c-1}]_{z^{2}=-3}\equiv 0 (mod 2^{h})$

.

(28)

Using this lemma, we obtain the following.
Theorem 5.4. Suppose that $a(c+1)$ -component link
is obtained from a -component
link by a coherent band surgery. If $P(L;i, i)=P(L’;i, i)=(-2)^{h}$ , then
$L’$

$c$

$L$

$[ \frac{\nabla(L;z)+z\nabla(L’;z)}{z^{c-1}}]_{z^{2}=-3}\equiv[\frac{\nabla(L;z)-z\nabla(L’;z)}{z^{c-1}}]_{z^{2}=-3}\equiv 0$$(mod 2^{h+1})$

.

(29)

114
6

The

$Q$

polynomialThepolynomial $Q(L;z)\in Z[z^{\pm 1}][3,11]$ is an invariant of the isotopy type of anunoriented link , which is defined by the following formulas:
$Q$

$L$

$Q(U;z)=1$ ;$Q(L_{+};z)+Q(L_{-};z)=z(Q(L_{0};z)+Q(L_{\infty};z))$ ,

where

$U$

is the unknot and

(30)
(31)

is an unoriented skein quadruple.

$(L_{+}, L_{-}, L_{0}, L_{\infty})$$)(\wedge\vee$$L_{-}$$L_{+}$$L_{0}$$L_{\infty}$

Figure 8: An unoriented skein quadruple.

Let $\rho(L)=Q(L;(\sqrt{5}-1)/2))$ . Then Jones [18] has shown(32)$\rho(L)=\pm\Gamma 5$

where $r=\dim H_{1}(\Sigma(L);Z_{5})$ .
Furthermore, Rong [39] has shown that there are six cases for the ratios amongas in Table 6.

$\rho(L_{+}),$$\rho(L_{0}),$$\rho(L_{-})$,$\rho(L_{\infty})$

Table 6: The values of thepolynomials at $z=(\sqrt{5}-1)/2.$$Q$

Using Table 6, we have criteria on the polynomials of two links which are related
by a crossing change [40, Theorem 4.1] or a band surgery [19, Theorem 3.1].
$Q$

Proposition 6.1. (i) If two links

$L$

and

$L’$

are related by a crossing change, then$\rho(L)/\rho(L’)\in\{\pm 1, -\sqrt{5}^{\pm 1}\}$

.

(33)

115
(ii) If two links$L$and $L’$ are related by a band surgery, then$\rho(L)/\rho(L’)\in\{\pm 1, \sqrt{5}^{\pm 1}\}$

.

(34)

Moreover, using Table 6, we have the following.

Theorem 6.2. Suppose that two links andare related by either a crossing change
or a band surgery and that $\rho(L)=\rho(L’)=\pm\Gamma 5$ . Then
$L$$\det L+\det L’\equiv 0$

$or$

$L’$

$\det L-\det L’\equiv 0$$(mod 5^{r+1})$

.

(35)

Example 6.3. $d_{cb}(9_{39}!, 6_{2}^{2})>1$ . Since$=\rho(6_{2}^{2})=-\sqrt{5},$ $\det(9_{39}!)=55$, and$\det(6_{2}^{2})=10$ , the result follows by
Theorem 6.2. Note that since $\sigma(9_{39}!)=2,$
we cannot use Proposition 2.2.
$\rho(9_{39}!)$$\sigma(6_{2}^{2})=3,$

7

Table of $d_{cb}(K, K’)$We give a table of coherent band-Gordian distances between two knots (cf: [15, Table 1])
Table 7: Coherent band-Gordian distances between two knots with up to 6 crossings.

$\dagger$: corrected

116Table 8: Coherent band-Gordian distances between two knots with up to 7 crossings.

The symbol2/4 means $d_{cb}(K, K’)=2$ or 4.

117
Acknowledgements
The first author was partially supported by KAKENHI, Grant-in-Aid for Scientific Research (C) (No. 21540092), Japan Society for the Promotion of Science.

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Department of Mathematics, Osaka City University
Sugimoto, Sumiyoshi-ku Osaka 558-8585
JAPAN-mail address: kanenobu@sci. osaka-cu. ac. jp$E$$*\beta RiF^{\infty}|\Delta_{arrow}\star\not\cong^{\backslash }\star\not\cong\beta_{\pi\Phi 5^{\backslash }ffl_{J\iota}^{i7}H}^{\Leftrightarrowt}\Leftrightarrow(_{\tau\supset}^{-}\equiv\ovalbox{\tt\small REJECT}^{B=},J^{\backslash }\Xi$

Osaka City University Advanced Mathematical InstituteSugimoto, Sumiyoshi-ku Osaka 558-8585
JAPAN-mail address: moriuchi@sci. osaka-cu. ac. jp$E$$\star\beta\varpi\pi_{\Delta 1_{arrow}}*\not\cong \mathscr{X}\not\cong\varpi f_{J\iota\overline{1^{J}}}^{n}H n^{*}h$EjE