38

数理解析研究所講究録
第2004巻 2016年 38-46

On handlebody‐links and \mathrm{M}\mathrm{i}\mathrm{l}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{s} link‐homotopy invariants
Yuka Kotorii

Graduate School of Mathematical Science, The University of Tokyo

Introduction

1

This is a survey of the joint work [13] with Atsuhiko Mizusawa.

A

handlebody‐link [11, 27] is a disjoint union of embeddings of handlebodies in the

3‐sphere S^{3} (Figure 1).

Two handlebody‐links areequivalent if there is an ambient iso‐

FIGURE 1. A handlebody‐link.

topy which transforms one to the other. An HL ‐homotopy is an equivalence relation on

handlebody‐links, which is analogous to link‐homotopy of links. Here, link‐homotopy is
generated by ambient isotopies and self‐crossing changes. In [22],Mizusawa and Nikkuni

showed that the HL‐homotopy classes of 2‐component handlebody‐links were classifiedcompletely by the linking numbers for handlebody‐links, which was defined by Mizu‐
sawa in [21]. In [13], we construct HL‐homotopy invariants for handlebody‐links by us‐
ing Milnors \overline{$\mu$}‐invariants for links. We then give a necessary and sufficient condition of

that

a

handlebody‐link is HL‐homotopic to a separable one by the extended Milnors\overline{$\mu$}‐invariants.

Here, a handlebody‐link is separable if there exists a disjoint union of 3‐balls such that each component of the handlebody‐link is contained in a distinct 3‐ball.

Moreover, we give a bijection between the set of HL‐homotopy classes of n ‐component

handlebody‐links with some assumption and a quotient of a tensor product of \mathb {Z} ‐modules
by the action of the general linear group.

39

Preliminaries

2

J. Milnor defined a family of invariants for an ordered oriented link in S^{3}[19, 20].

eralization of the linking numbers, in

as

a

gen‐

These invariants are called Milnors \overline{ $\mu$}-invariants. For an ordered oriented n ‐component link L , Milnors \overline{$\mu$}‐invariant is specified

by a sequence I of indices in \{ 1, 2,

.

.

.

,

n\} and denoted by \overline{ $\mu$}_{L}(I)

If the sequence is with.

distinct indices, then this invariant is also link‐homotopy invariant and called Milnorslink‐homotopy invariant.
We introduce the definition of Milnors link‐homotopy invariants, and to give invariants

for handlebody‐links, we show that these are additive under a bund sum for components.
Let L=L_{1}\cup\cdots\cup L_{n} be an ordered oriented n ‐component link in S^{3}

Consider the link.

$\pi$=$\pi$_{1}(S^{3}\backslash L_{1}\cup\cdots\cup L_{n-1}) of L_{1}\cup\cdots\cup L_{n-1} and denote the i‐th meridian by m_{i}groupfor i(1\leq i\leq n-1).

Given a finitely generated group G the reduced group \overline{G} is defined to the quotient of G,

by its normal subgroup generated by [g, hgh^{-1}] for any g, h\in Gcommutator of a and b\mathbb{Z}[[X_{1}, . . . , X_{n-1}]]

Let

X_{1}

,

.

.

.

.

,

X_{n-1},[a, b] means thewhere

Then \overline{ $\pi$} is generated by the meridians m_{1}, m_{2} ,

.

.

.

,

m_{n-1}.

be the non‐commutative formal power series ring generated by

Denote by \hat{Z} its quotient ring by the two‐side ideal generated by all mono‐.

mials in which at least one of the generators appear at least twice. The Magnus expansion

homomorphism from the free group F(m_{1}, \ldots, m_{n-1}) generated by m_{1}$\varphi$ is

a

into

\mathbb{Z}[[X_{1}, . . . , X_{n-1}]] defined by sending m_{i} to 1+X_{i} and m_{i}^{-1} to 1-X_{i}+X_{i}^{2}-\cdots

,

.

.

,

m_{n-1},

induces a homomorphism from.

\overline{F(m_{1},\ldots,m_{n-1})} into \hat{Z} Let w_{n}\in F(m_{1}, . . . , m_{n-1}) be.

word representing L_{n} in \overline{ $\pi$} We then define $\mu$_{L}(i_{1}i_{2}\ldots i_{r}n) for distinct indices i_{1}, i_{2} ,
.

as

.

.

.

.

,

It
a

i_{r}, nthe coefficient of the Magnus expansion of w_{n} in \hat{Z} :$\varphi$(w_{n})=1+\displaystyle \sum$\mu$_{L}(i_{1}i_{2}\ldots i_{r}n)X_{i_{1} X_{i_{2} \ldots X_{i_{r} ,where the summation is over all sequences i_{1}i_{2}\ldots i_{r} with distinct indices between 1 andn-1.

Similarly, we define $\mu$_{L}(i_{1}i_{2}\ldots i_{s}) for any distinct indices between 1 and n.

We

the residue class of $\mu$_{L}(i_{1}i_{2}\ldots i_{r}n) modulo theindeterminacy\overline{ $\mu$}_{L}(i_{1}i_{2}\ldots i_{r}n) as\triangle_{L}(i_{1}i_{2}\ldots i_{r}n) which is the greatest common divisor of $\mu$_{L}(j_{1}j_{2}\ldots j_{s})' \mathrm{s} where j_{1}j_{2}\ldots j_{s}define,rangesover

all sequences obtained by deleting at least one of the indices i_{1}, i_{2}

,

.

.

.

,

i_{r}, nand permuting the remaining ones cyclicly.

Moreover we define \triangle_{L}(i_{1}n)= O.

to this, for any

define \overline{ $\mu$}_{L}(I) for any sequence I of distinct

indices in \{ 1,2,

Theorem 2.1

n

.

.

‐component link L.

,,

we canSimilarn\}([19, 20

If L and L' are link‐homotopic, then \overline{ $\mu$}_{L}(I)=\overline{ $\mu$}_{L},(I) for any

sequence I with distinct indices.

40([20]). Let L be an ordered oriented link. Then the following relations hold.
(1) \overline{ $\mu$}_{L}(i_{1}i_{2}\ldots i_{m})=\overline{ $\mu$}_{L}(i2. . . i_{m}i_{1})(2) If the orientation of the k‐th component of L is reversed, then \overline{ $\mu$}_{L}(i_{1}i_{2}\ldots i_{m}) is multi‐

Lemma 2.2

plied by+1 or-1 according as the sequence i_{1}i_{2}\ldots i_{m} contains k an even or odd numberof times.
The following lemma is used for Proposition 3.4. This lemma is showed by using thedefinition of Milnors link‐homotopy invariants.
Lemma 2.3. Let L=L_{1}\cup L_{2}\cup\cdots\cup L_{n-1} be an (n-1) ‐component link in S^{3}

K' be disjoint knots in S^{3}\backslash L.

Let K and.

Let I be a sequence with distinct indices in \{ 1, 2,

.

.

.

,

n\}.

If I contains the index n,$\mu$ L\cup(K_{b}\# K')(I)\equiv$\mu$_{L\cup K}(I)+$\mu$_{L\cup K'}(I)where K\#_{b}K' isaband sum

L\cup K and L\cup K'

are n

mod

\mathrm{g}\mathrm{c}\mathrm{d}(\triangle_{L\cup K}(I), \triangle_{L\cup K'}(I)of K and K' with respect to any band,,and L\cup(K\#_{b}K') ,‐component links whose n‐th components are K\#_{b}K', K and K',

respectively.
Remark 2.4.
sum

By a property of the \overline{$\mu$}‐invariant, we can obtain the same result for a bandof the i‐th component instead of the n‐th component.

Remark 2.5. In[14], V. S. Krushkal showed Milnors \overline{$\mu$}‐invariants are additive underconnected sum for links which are separated by a 2‐sphere.

\mathrm{M}\mathrm{i}\mathrm{l}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{s}\overline{ $\mu$}‐invarinats for handlebody‐links

3

In this section, we define the HL‐homotopy, which is an equivalence relation on handlebody‐links and construct HL‐homotopy invariants for handlebody‐links by using Milnors \overline{ $\mu$}invariants.

Definition 3.1(HL‐homotopy). Let H_{0} be n handlebodies and H_{i}(i=1,2) two n‐component handlebody‐links obtained by embedding H_{0} to S^{3} by f_{i}.

Two handlebody‐links H_{1} and H_{2} are called HL ‐homotopic if there is homotopy h_{t} from f_{1} to f_{2} where thecomponents of h_{t}(H_{0}) are mutually disjoint at any 0\leq t\leq 1.
Remark 3.2. In

troduced.

[22], the notation of neighborhood homotopy of spatial graphs was in‐

A spatial graph is an embedding of graph in S^{3}.

We can represent the HL‐

homotopy of handlebody‐links by the neighborhood homotopy of spatial graphs.
Let H=L_{1}\cup\cdots\cup L_{n} be an n ‐component handlebody‐link with genus g_{i} for each i.
Let\{e_{1}^{i}, . . . , e_{g}^{i_{i}}\} beabasis of H_{1}(L_{i};\mathbb{Z}) and\mathcal{B}=\{e_{1}^{1}, . . . , e_{g_{1}}^{1}, . . . , e_{1}^{n}, . . . , e_{g_{n}}^{n}\}.

We can41

regard an element of B as an embedded closed oriented circle in S^{3} So the disjoint union.

e_{k_{1} ^{1}\cup e_{k_{2} ^{2}\cup\cdots\cup e_{k_{n} ^{n}canbe regarded as an ordered oriented link for each k_{i}(1\leq k_{i}\leq g_{i})Let I be a sequence of length m(m\leq n) with distinct indices in \{ 1, 2,

.

.

.

,

n\}

.

.

For each

I , we define an element M_{H,\mathcal{B}}(I) of tensor product space (\mathbb{Z}/\triangle_{I}\mathbb{Z})^{g_{1} \otimes\cdots\otimes(\mathbb{Z}/\triangle_{I}\mathbb{Z})^{g_{n} as\mathbb{Z}/\triangle_{I}\mathbb{Z}‐module defined bywhere

M_{H,B}(I):=\displaystyle\sum_{k_{1},.k_{n}=1}^{g_{1}.'\cdot,g_{n}\overline{$\mu$}_{e_{k_{1}^{1}\cup\cdots\cupe_{k_{n}^{n}(I)e_{k_{1}^{1}\otimes\cdots\otimese_{k_{n}^{n},divisor of all\triangle_{e_{k_{1} ^{1}\cup\cdots\cup e_{k_{n} ^{n} (I)\overline{ $\mu$}_{e_{k_{1} ^{1}\cup\cdots\cup e_{k_{n} ^{n} (I) is in \mathbb{Z}/\triangle_{I}\mathbb{Z}, \triangle_{I} is the greatest
k_{n} where \triangle_{e_{k_{1} ^{1}\cup\cdots\cup e_{k_{n} ^{n} (I) is indeterminacy of the original Milnors invariant

for all k_{1}common,.

.

.

for the link,,

e_{k_{1} ^{1}\cup e_{k_{2} ^{2}\cup\cdots\cup e_{k_{n} ^{n} and e_{k_{i} ^{i} is the canonical basis(\mathbb{Z}/\triangle_{I}\mathbb{Z})^{9i} as \mathbb{Z}/\triangle_{I}\mathbb{Z} ‐module.

(0, \ldots, 0,\check{1}, 0, \ldots, 0)k_{i}ofRemark 3.3. If the first homology group of each component of H is \mathb {Z} the M_{H,B}(I) is,

identified with the original Milnors link‐homotopy invariant for a link, essentially.

(\mathbb{Z}/\triangle_{I}\mathbb{Z})^{g_{1} \otimes\cdots\otimes(\mathbb{Z}/\triangle_{I}\mathbb{Z})^{g_{n} and denote by M_{H}(I) the residue class of M_{H,I3}(I) by the action for (\mathbb{Z}/\triangle_{I}\mathbb{Z})^{g_{1} \otimes\otimes(\mathbb{Z}/\triangle_{I}\mathbb{Z})^{g_{n} .
We consider

.

.

a

natural action of GL(g_{1}, \mathbb{Z})\times\cdots\times GL(g_{n)}\mathbb{Z})on.

Proposition 3.4. Let H be an n ‐component handlebody‐link. Then M_{H}(I) is independentof a basis \mathcal{B} of H_{1}(H, \mathbb{Z}) and an HL ‐homotopy invariant.
Proof. The proof is by induction on the length m of sequence I.

We can show it by using

properties of \overline{$\mu$}‐invariants for links (Lemma 2.2 and 2.3). See [13] for details.

\squareExample 3.5. Let H be a handlebody‐link which are the regular neighborhood of graphillustrated in

Figure 2.

\triangle_{e_{k_{1} ^{1}\cup e_{k_{2} ^{2}\cup e_{k_{3} ^{3} (I)=0

Let I=123

.

Then,

in other cases. So \triangle_{I}=2 and\triangle_{e_{1}^{1}\cup e_{1}^{2}\cup e_{1}^{3} (I)=\triangle_{e_{1}^{1}\cup e_{1}^{2}\cup e_{2}^{3} (I)=2 and

M_{H}(I)=1e_{1}^{1}\otimes e_{1}^{2}\otimes e_{1}^{3}+1e_{2}^{1}\otimes e_{2}^{2}\otimes e_{2}^{3}\in(\mathbb{Z}_{2})^{2}\otimes(\mathbb{Z}_{2})^{2}\otimes(\mathbb{Z}_{2})^{2}
We can show the following corollary by using clasper theory introduced by Habiro [8].

Corollary 3.6. An n ‐component handlebody‐link H is HL ‐homotopic to a separable handlebody‐link if and only if M_{H}(I)=0 for any I.

Remark 3.7. T.

Fleming defined a numerical invariant $\lambda$_{ $\Phi$}(H) of a pair of a spatial

graph $\Phi$ and its subgraph H under component homotopy in[3]. Now, we define $\Phi$ as ahandlebody‐link instead of a spatial graph and H as its component instead of a subgraph.
We then can naturally extend this invariant to a pair of a handlebody‐link and its com‐ponent under HL‐homotopy. Then, the value of $\lambda$_{ $\Phi$}(H) is the length of first non‐vanishingfor M_{ $\Phi$}(I) such that I contains the component number of H.

42

FIGURE 2.

4

Handlebody‐link H.

Main Theorem
Let \mathbb{H}[g_{1}, g_{2}, \cdots, g_{n}] be the set of n ‐component handlebody‐links with genus g_{i} for each1\leq i\leq n such that its any (n-1) ‐component subhandlebody‐link is HL‐homotopic to a

separable handlebody‐link. By Corollary 3.6, this condition is equivalent to that its any

M(I) s of length less than n vanishes.
Let S be a permutation group on \{2, 3, . . . , n-1\}
I_{ $\sigma$} as a sequence 1 $\sigma$(23\cdots n-1)n.
Theorem 4.1. For any element $\sigma$ in S,.

For any element a in S , we define

the map$\varphi$:\mathbb{H}[g_{1}, \cdots , g_{n}]\rightar ow\oplus_{ $\sigma$\in S}(\mathbb{Z}^{g_{1} \otimes\cdots\otimes \mathbb{Z}^{g_{n} )H\mapsto(M_{H}(I_{ $\sigma$}))_{ $\sigma$\in S}
inducesabijection between the set of HL ‐homotopy classes of \mathbb{H}[g_{1}, g_{2}, \cdots, g_{n}]residue class of\oplus_{ $\sigma$\in S}(\mathbb{Z}^{g_{1} \otimes\cdots\otimes \mathbb{Z}^{g_{n} )and theby diagonal action of general linear group.

We give two examples.

Example 4.2. Let I=123.

Let

H_{1} and H_{2} be two handlebody‐links which are theregular neighborhood of graphs depicted in Figure 3. Then, \triangle_{I}=0 and

M_{H_{1}}(I)=1e_{1}^{1}\otimes e_{1}^{2}\otimes e_{1}^{3}+1e_{1}^{1}\otimes e_{2}^{2}\otimes e_{1}^{3}+1e_{1}^{1}\otimes e_{3}^{2}\otimes e_{1}^{3}+2e_{1}^{1}\otimes e_{1}^{2}\otimes e_{2}^{3}+2e_{1}^{1}\otimes e_{2}^{2}\otimes e_{2}^{3}+2e_{1}^{1}\otimes e_{3}^{2}\otimes e_{2}^{3}\in \mathbb{Z}^{2}\otimes \mathbb{Z}^{3}\otimes \mathbb{Z}^{2}.

M_{H_{2}}(I)=1e_{1}^{1}\otimes e_{1}^{2}\otimes e_{1}^{3}+1e_{1}^{1}\otimes e_{2}^{2}\otimes e_{1}^{3}+1e_{2}^{1}\otimes e_{1}^{2}\otimes e_{1}^{3}+1e_{2}^{1}\otimes e_{2}^{2}\otimes e_{1}^{3}
1 e_{1}^{1}\otimes e_{1}^{2}\otimes e_{2}^{3}+1e_{1}^{1}\otimes e_{2}^{2}\otimes e_{2}^{3}+1e_{2}^{1}\otimes e_{1}^{2}\otimes e_{2}^{3}+1e_{2}^{1}\otimes e_{2}^{2}\otimes e_{2}^{3}\in \mathbb{Z}^{2}\otimes \mathbb{Z}^{3}\otimes \mathbb{Z}^{2}.
We have that M_{H_{1}}(I) is transformed to M_{H_{2}}(I) by the diagonal action of general linear
group. Therefore H_{1} and H_{2} are HL‐homotopic.

43

FIGURE 3.

Example 4.3. Let I=123

Handlebody‐links H_{1} and H_{2}.

Let H3 and H_{4} be two handlebody‐links which are theregular neighborhood of graphs depicted in Figure 4. Then, \triangle_{I}=0 and.

M_{H_{3}}(I)=1e_{1}^{1}\otimes e_{1}^{2}\otimes e_{1}^{3}+1e_{1}^{1}\otimes e_{2}^{2}\otimes e_{1}^{3}+1e_{2}^{1}\otimes e_{1}^{2}\otimes e_{1}^{3}+1e_{2}^{1}\otimes e_{2}^{2}\otimes e_{1}^{3}+1e_{1}^{1}\otimes e_{3}^{2}\otimes e_{2}^{3}+1e_{2}^{1}\otimes e_{3}^{2}\otimes e_{2}^{3}\in \mathbb{Z}^{2}\otimes \mathbb{Z}^{3}\otimes \mathbb{Z}^{2}.

M_{H_{4}}(I)=2e_{1}^{1}\otimes e_{1}^{2}\otimes e_{1}^{3}+2e_{2}^{1}\otimes e_{1}^{2}\otimes e_{1}^{3}+1e_{1}^{1}\otimes e_{2}^{2}\otimes e_{2}^{3}+1e_{2}^{1}\otimes e_{2}^{2}\otimes e_{2}^{3}\in \mathbb{Z}^{2}\otimes \mathbb{Z}^{3}\otimes \mathbb{Z}^{2}.
We can show that H_{1} is not HL‐homotopic to H_{2} by using some invariants for the actionof general linear group\mathrm{o}\mathrm{n}_3\mathrm{t}\mathrm{h}\mathrm{e}tensor product space. See[13]\mathrm{f}\mathrm{o}\mathrm{r}3

H_{3} :FIGURE 4.

Handlebody‐links H3 and H_{4}.

details.

44

Acknowledgements
The author would like to thank Professor Tomotada Ohtsuki for inviting me the work‐

shop



Intelligence of Low‐dimensional Topology 2016

She would also like to thankProfessor Sadayoshi Kojima and Professor Mitsuhiko Takasawa for your advice.

References[1] A. Cayley. On the theory of linear transformations. Cambridge Math. J., 4:

193‐

209, 1854.

[2] A. Cayley. On the theory of determinants. Trans. Cambridge Philos. Soc., 8, no. 7:
75‐88, 1849.

[3] T. Fleming. Milnor invariants for spatial graphs. Topology Appl. 155 (2008), no. 12,
1297‐1305.

[4] T. Fleming, A. Yasuhara, Milnor
Math. Soc. 137 (2009) 761‐770.

sinvariants and self C_{k} ‐equivalence, Proc. Amer.

[5] M.N. Gusarov, Variations of knotted graphs. The geometric technique of n‐equivalence. (Russian), Algebra i Analiz 12 (2000), no. 4, 79−125; translation in
St. Petersburg Math. J. 12 (2001), no. 4, 569‐604.
[6] N. Habegger and X.‐S. Lin, The classification of links up to link‐homotopy, J. Amer.
Math. Soc. 3:2 (1990), 389‐419.
[7] K. Habiro, Clasp‐pass moves on knots, unpublished, 1993.
[8] K. Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000), 1‐83.
[9] F. L. Hitchcock. The expression of a tensor or a polyadic as a sum of products. J.
Math. Phys., 6(1): 164‐189, 1927.

[10] F. L. Hitchcock, Multiple invariants and generalized rank of a p ‐way matrix or tensor.
J. Math. Phys., 7(1): 39‐79, 1927.
[11] A. Ishii, Moves and invariants for knotted handlebodies, Algebr. Geom. Topol. 8(2008), 1403‐1418.

[12] K. Johannson, Topology and combinatorics of 3‐manifolds, Lecture Notes in Mathe‐
matics 1599, (1995) Springer‐Verlag, Berlin.
[13] Y. Kotorii and A. Mizusawa, HL ‐homotopy of handlebody‐links and Milnor
ants, arXiv:math /1603.09067.

sinvari‐45[14] V. S. Krushkal, Additivity properties of Milnor s\overline{ $\mu$} ‐invariants, J. Knot Theory Ram‐
ifications 7 (1998), no. 5, 625‐637.
[15] K. Makino and S. Suzuki, Notes on neighborhood congruence of spatial graphs, Gaku‐
jyutu Kenkyu, School of Education, Waseda Univ., Ser. Math., 43 (1995), 15‐20.

[16] S. V. Matveev, Generalized surgeries of three‐dimensional manifolds and representa‐
tions of homology spheres (Russian), Mat. Zametki 42 (1987), 268‐278, 345.
[17] J‐B. Mailhan, invariants de type fini des cylindres dhomotogie et des string links,
Thèse de Doctorat (2003), Université de Nantes.
[1S] J‐B. Meilhan, Y. Yasuhara, On C_{n} ‐moves for links. Pacific J. Math. 238 (2008), no.
1, 119‐143.

[19] J. Milnor, Link groups, Annals of Mathematics (2), 59 (1954), p177‐195.
[20] J. Milnor, Isotopy of links, Algebraic geometry and topology, A symposium in honorof S. Lefschetz, pp. 280‐306, Princeton University Press, Princeton, N. J., 1957.

[21] A. Mizusawa, Linking numbers for handlebody‐links, Proc. Japan Acad. Ser. A Math.
Sci. 89 (2013), 60‐62.
[22] A. Mizusawa and R. Nikkuni A homotopy classification of two‐component spatial
graphs up to neighborhood equivalence, Topology Appl. 196 (2015), part B, 710‐718.
[23] T. Motohashi and K. Taniyama, Delta unknotting operation and vertex homotopy ofgraphs in R^{3} KNOTS 96 (Tokyo), 185‐200, World Sci. Publ., River Edge, NJ, 1997.
,[24] H. Murakami and Y. Nakanishi, On a certain move generating link‐homology, Math.
Ann. 284 (1989), 75‐89.

[25] T. Soma, H. Sugai and A. Yasuhara, Disk/band surfaces of spatial graphs. Tokyo J.
Math. 20 (1997), 1‐11.

[26] S. Suzuki, Local knots of 2‐spheres in 4‐manifolds, Proc. Japan Acad. 45 (1969),
34‐38.

[27] S. Suzuki, On linear graphs in 3‐sphere, Osaka J. Math. 7 (1970), 375‐396.
[28] S. Suzuki, On surfaces in 3‐sphere: prime decompositions, Hokkaido Math. J. 4
(1975), 179‐195.
[29] K. Taniyama and A. Yasuhara, Clasp‐pass moves on knots, links and spatial graphs,
Topology Appl. 122 (2002), 501‐529.

46[30] A. Yasuhara, Self Delta‐equivalence for Links Whose Milnors Isotopy Invariants
Vanish, Trans. Amer. Math. Soc. 361 (2009), 4721‐4749.
Graduate School of Mathematical Science
The University of Tokyo

Tokyo 153‐8914
JAPAN\mathrm{E} ‐mail address:

kotorii@ms.u‐tokyo.ac.jp\mathrm{R}\overline{\mathrm{g}\mathrm{p}_{\backslash} $\lambda$\mp$\lambda$\mp\mapsto$\beta$_{$\pi$\mathscr{X}\ovalbox{\t\smal REJECT}\ovalbox{\t\smal REJECT}_{\backslash}^{\backslash},\}^{\mathrm{R} ^{\mathrm{A} \neq\mathrm{f}\mathrm{f}\mathrm{l}_{J^{\mathrm{D} ^{$\nu$_{\mathrm{u} 7^{\backslash}\backslash\}\rflo r\backslash\ovalbox{\t\smal REJECT}_{\backslash}\mathrm{E}\mathrm{f}\mathrm{f}\mathrm{i}\ovalbox{\t\smal REJECT}