数理解析研究所講究録 1374 巻 2004 年 179-187

173

格子から切り取った平面曲線と Dehn 手術の係数(subtitle: Plane slalom curves of a certain type,pretzel links and Kirby-Melvin’s Grapes)
山田裕一 (Yuichi YAMADA)
電気通信大学 (The Univ. of ElectrO-Communications)

March 14, 2004
Abstract

We are concerned with plane curves of type $C(p, q, r)$ as in Figure 1 and 2, andtheir corresponding links $L(C(p, q, r))$ via A’Campo’s divide theory, where , ,are positive integers with $1\leq p\leq q\leq r.$ We will point out that 2-fold covering
spaces of the 3-dimensional spherebranched along $L(C(p, q, r))$ (2-branchedcoverings, for short) is represented by Kirby-Melvin’s grapes. We will also refer
to some other related topics.
$p$

$q$

$r$

$S^{3}$

1Introduction.
The divide is a relative, generic immersion of a1-manifold in a unit disk $D$ inN. A’Campo formulated the way to associate to each divide
a link $L(C)$ in theA2,([Al,A3,A4]):
3-dimensional sphere$\mathrm{R}^{2}$$C$$\mathrm{S}^{3}$$\mathrm{L}(\mathrm{C})=\{(u, v)\in TD|u\in C, v\in TUC, |u|^{2}+|v|^{2}=1\}$$\subset S^{3}$.

The class of links of divides properly contains the class of the links arising from isolated
singularities of complex curves. In this paper, we draw only curves
but the disk.
, whereNote that the number of components of $L(C)$ is(and, respectively) is the number of immersed components of arcs (and circles) in .
We say that is in arc case if $\beta_{a}(C)=1$ and.
, Secondary.
02000 Mathematics Subject Classification: Primary$C$$\mathfrak{g}_{a}(C)+2\Downarrow_{c}(C)$$\mathrm{L}(\mathrm{C})$$C$$\mathfrak{g}_{c}(C)$$C$$\mathrm{L}(\mathrm{C})=0$$57\mathrm{M}25,14\mathrm{H}20$$55\mathrm{A}25$

Keywords: Pretzel knots, plane curves, branched coverings, framed links
xThis work is partialy supported by Grant-in-Aid for Scientific Research N0.15740034, Japan Society
for the promotion of Science.
Here the author focus on a part of (not whole) his talk on NOv.28, 2003. Thus he has add a subtitleof this article as above.

180$\mathrm{q}$$\ldotsarrow \mathrm{q}$$\mathrm{X}$$\ldots$orへ
Figure 1: $C(p, q, r)$ and $D(p, q, r)$

Figure 2: $C(2,3,7)$ ,$C(3,3,5)$

and$\mathrm{C}(\mathrm{p}, 4,6)$

For a plane curve of type $C(p, q, r)$ , by $D(p, q, r)$ , we denote the corresponding
diagram in Figure 1. Numbers in the diagram are written only for counting. Note thateach odd number in $\{p, q, r\}$ corresponds to white point and ” at the terminal, andare given by;andthat$” \mathrm{a}$$\#_{a}(C)$$\Downarrow_{c}(C)$$\#_{a}(C(p, q, r))=e(p, q, r)$,$\#_{c}(C(p, q, r))=\{\begin{array}{l}1\mathrm{i}\mathrm{f}e(p,q,r)=00\mathrm{i}\mathrm{f}e(p,q,r)\geq 1\end{array}$where $e(p, q, r)$ is the number of even number(s) in $\{p, q, r\}$ . Curves $C(p, q, r)$ in arc cases
are included in the class of slalom curves, which was studied by N. A’Campo in [A2]
(Theorem 4.1 in Section. 4 is one of his results). We will study the links $L(C(p, q, r))$from mainly the point of view of 4-dimensional topology, branched coverings, KirbyMelvin’s grapes and moves of framed links.
The author would like to sincere gratitude to Professor N. A’Campo for his kindencouragement by -mail. The author would like to thank to Professor Masaharu) on A’Campo’s theory and to Professor
Ishikawa for many valuable advice ([Gil,and checkedMikami Hirasawa, who informed him the starting example
some examples of Theorem 2.1 by more knot-theoretical and visualized method in [H].
The author would like also to thank knot theorists Prof. Koya Shimokawa, Dr. KazuhiroIchihara and Dr. Takuji Nakamura for their valuable comments from their own recent
research.
$\mathrm{e}$$\mathrm{G}\mathrm{I}2]$$\mathrm{P}\mathrm{r}(-2,3,7)$

181

Figure 3:

Blow-down$\Leftrightarrow$

Figure 4:

2

A triangle move

Pretzel links

First, we give an answer to the question “what link is $L(C(p, q, r))?$ ”

Theorem 2.1 The link $L(C(p, q, r))$ is a pretzel link of type $(-1, p, q, r)$ .

. In the small cases $(p, q, r)=(1,1,1)$ , (1, 1, 2), (1, 2, 2) and (2, 2, 2), it is easily$Pro\mathrm{o}/$checked by the standard singularity theory, or by [H]. In fact, the link is:a Hopflink,:a trefoil knot, :a torus link
or :a torus link $T(3,3)$ , respectively. In
general cases, it is proved by some blow-down’s, i.e. full-twistings, see Figure 3. Notethat one blow-down increase one of , , by two.
$A_{1}$$A_{2}$$A_{3}$$T(\mathit{2},\mathit{4})$

$p$

$q$

$D_{4}$$\square$$r$

2-branched coverings of along such pretzel links are known to be Seifert manifolds.
Akbulut-Kirby’s algorithm [AK] is useful.
$5^{3}$

Corollary 2.2 Tie 2-fold covering space $M^{3}(p, q, r)$ ofalong $L(C(p, q, r))$ is a$\{-1;(0,0);(p,1),(q,1),(r,1)\}$
Seifert manifold of typein Orlik’s notation].
$57^{3}$$\int Or$The 3-manifold $M(p, q, r)$ (as a boundary of the 4-manifold( , , )) is represented
by a framed link in Figure 5 where every framing is -2, thus omitted. Note thatthe 4-manifold $W(p, q, r)$ directly corresponds to the diagram $D(p, q, r)$ , see [HKK,p.13 and 25]. Such special framed links are represented by Kirby-Melvin’s useful
method “grapes” [KM]: A grapes is a configuration of hexagonally packed circles. Eachindividual circle will be called a grape. For the way to construct from a grapes to its
framed link and more detail, the author strongly recommend to the readers to see [KM].
The advantage of representation by grapes is slip of a grape, i.e., that we can move agrape under a certain conditions without changing of the 4-manifold. On the other
hand, by similar Kirby calculus to that in [ , p.15], it is proved that $M(p, q, r)$ is also
represented by a framed link in Figure 7 (of course, $W$ ( , , ) has been changed).
$W^{4}$$\mathrm{K}$

$p$

$q$

$r$$p$

$q$

$r$

182

$\cap \mathrm{O}$$\cap \mathrm{O}$$\cap\cdot$$\mathrm{O}\Gamma J\Gamma I\Gamma J_{\wedge^{1}}\cap$$\mathrm{O}\Gamma y\Gamma y\Gamma y\Gamma y\bigcap_{\vee}\mathrm{Q}^{1}$$\check{.}\mathrm{O}$

Figure 5:

Figure 6:

Framed links and Grapes

Triangle Moves and Grape Slips from Fig.5

183$\mathrm{r}$$0_{\vee}^{1}\cap \mathrm{o}_{9}^{\mathrm{p}}0^{[}$

Figure 7:$M(p, q, r>)$In divide theory, triangle moves on divides in Figure 4 do not change the corresponding links 2. See the moves from Figure 5 to Figure 6 (and see [AGV, p.117]).
There might be a relationship between triangle moves and slips of grapes, but maybe
indirectly, since the former is local and the latter is global.

3Triangle singularities

Each Seifert manifold of type $\{-1; (0,0);(p, 1), (\mathrm{g}, 1), (r, 1)\}$ for 14 triples $(p, q, r)$ in
Table 1 is known to be a link of Arnold’s triangle singularities ([Ar]) (exceptional, i.e., an intersection of the complex
singularities or unimodal singularities)
inalgebraic surface and a small 5-sphere centered at the singularity. Here we copy the list
as Table 1 from [ , p.63] (see [Ar], [AGV, p.110] and also [Mz]).
$D_{p,q,r}$$\mathrm{C}^{3}$$\mathrm{D}$Question 1. Are there any topological or algebraic-geometrical relationship between
the plane curves $C(p, q, r)$ and the singularities?
$D_{p,q,r}$Table 1: List of Triangle singularities, Prof. Masaharu Ishikawa has pointed out that the converse is not true and given infinitely].
many examples of different types after earlier discoveries in [Gil,$2\mathrm{R}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{l}\mathrm{y}$$\mathrm{G}\mathrm{I}2$

184
There is a symmetry called “Arnold’s strange duality” between Dolgachev numbers$(p, q, r)$ and Gabrielov numbers $(p’, q’, r’)$ in the list. The resolution space of the singularity
is orientation-reversingly 3 diffeomorphic to the 4-manifold described by
the framed link in Figure 7.
$D_{p,q,r}$4

Related studies

Here we refer to some related works.
,We start with knot theory on , , -singularity. The link ofandsingularity inis torus knot of type $(2, 2k +1)$ , $(3, 4)$ and $(3, 5)$ respectively. For]
divide theory on torus links $T(a, b)$ (singularities of type $z^{a}-w^{b}=0$ ), see [AGV,$n\geq
4,$
and [GHY]. For
$A$

$D$

$E$

$\mathrm{A}2\mathrm{j}\mathrm{b}$$E_{6}$$E_{8}$$\mathrm{C}^{2}$$\mathrm{G}\mathrm{Z}$$L(C(2,2,n-2))$$=\mathrm{P}\mathrm{r}(-1,2,2, n-2)=\mathrm{P}\mathrm{r}(-2,2, n-2)$(Dn).

singularity $x^{2}+y^{3}+z^{3}=0$ inis the 2-branched covering of 573 along$L(C(2,2,2))$ ( Torus link $T(3,3)$ ) is a quotient space ofby the quaternion group. In [Y1], we studied a certain surgeryof order 8, called “quarternionic space”
along , from the view point of 4-manifold theory. On E-singularities,The link of$\mathrm{C}^{3}$$D_{4}$$5^{3}$$=$$Q_{8}$$G_{8}$$Q_{8}$

T(a, 4)$L(C(2,3,5))$ $=\mathrm{P}\mathrm{r}(-2,3,5)=$ T(a, 5)$L(C(2,3,3))$$=$$\mathrm{P}\mathrm{r}(-2,3,3)$(E6),$=$(Es).

Next, we study the links from the view point of Dehn surgery on hyperbolic knots.
A curve of type $C(2,3, n)$ with $n\geq 5$ is moved by triangle moves as in Figure 8 $(n=7$case). These three curves are obtained by “cutting out from a lattice $X$ ” as $X\cap$ ,where I is a union of rectangles in the plane. In [Y2, Y3] we pointed out that, in, the area I is related to coefficient of finite Dehn surgery,such curves of type $X$
i.e. surgery yielding 3-manifolds whose fundamental group is finite. Mainly hyperbolic
knots have been researched ([CGLS] and many works). From such a view point, thefollowing result by N. A’Campo is important:$\mathrm{I}$”$J?$

Theorem 4.1 ([A2]) For a slalom curves in arc cases, if the corresponding diagram isneither Dynkin diagram of typewith $k\geq 1$ ,nor , then the corresponding
divide knot is hyperbolic.
$A_{2k}$$E_{6}$$E_{8}$In the triangle moves in Figure 8, the area of I changes from $2n+8$ to $2n+6$ and to$2n+5.$ Koya Shimokawa and Kazuhiro Ichihara pointed out to the author that thesenumbers are near the special numbers (slopes) of the knots $Pr(-2,3,n)$ with odd $n\geq 7$
for Dehn surgery and informed M. Dunfield’s program to calculate boudary slopes of theknots. See Table 2. The data in the first four lines were picked up from K. Shimokowa’s
OHP-sheat used in his talk in Kobe, in Sep. 2003.
$3\mathrm{T}\mathrm{h}\mathrm{e}$author’s orientation may b opposite from the ordinary one here.
$\mathrm{e}$185

Figure 8:

area of$A’(\Re)$ below
$\Re$

$—$

Triangle moves as curves in the lattice

$—$

–

–

$\mathrm{o}$$\mathrm{o}$$\mathrm{o}$–$\circ$$\circ$$\circ$––Table 2: Special slopes for $Pr(-2,3, n)$whereFin.

$=$lens$=$Seif.

$=$$\mathrm{T}\mathrm{o}\mathrm{r}$.

Bdr.

$=$

$=$

finite (but non-cyclic) surgery, i.e.,yielding a 3-manifold whose fundamental group is non-cyclic finite,yielding a lens space,(ex. 19-surgery on $Pr(-2,3,7)$ is $L(19,8)$ , see [FS] and also [Y2])yielding a Seifert manifold,,toroidal surgery,yielding a 3-manifold that contains an essential torus,
boundary slope, i.e.,there exists an essential surface in the knot exterior
whose boundary curves has the slope,$\mathrm{i}.\mathrm{e}.$but we do not refer to these terminologies in detail here.
Back to Figure 8 again, we set$A’(\mathrm{I}):=$(the area of t) –“the number ofThen, in the triangle moves,$\mathrm{A}’(\%)$$270^{\mathrm{o}}$-corner (i.e. concave ones)”.

changes from $2nl$$6$

to $2n+5$ and to $2n+4.$

188
for the curves or the corresponding knots for
Question 2. Does the number
general I have mathematical meanings ?
$A’(\mathrm{I})$

Finally, we give one more information from knot theory. Any divide knot is known to
be a closure of strongly quasi-positive braid, i.e., of a composite of special conjugationof positive generators. Takuji Nakamura pointed out that any $Pr(-1,p, q)r)$ with, , $r>0$ is a closure of positive braid, i.e. of a composite of positive generators, ofindex 3. According to the author’s knowledge ([Yl, Y2], and [B]), it seems that any
divide knot yielding finite surgery is a closure of positive braid. It seems also that anyknot yielding lens spaces is a closure of positive braid, of course up to mirror image.

$p$

$q$

References[A1] N. A’Campo, Generic immersion of curves, knots, monodromy and gordian number, Inst.Hautes
Etudes Sci. Publ. Math. 88 (1998) 151-169.
[A2] N. A’Campo, Planar trees, slalom curves and hyperbolic knots, Inst.Hautes Etudes Sci.
Publ. Math. 88 (1998) 171-180.

[A3] N. A’Campo, Real defor mations and complex topology of plane curve singularities, Ann. de laFaculte des Sciences de Toulouse 8 (1999) 5-23.
[A4] N. A’Campo, Quadratic vanishing cycles, reduction curves and reduction of the monodromy groupof plane curve singularities, Tohoku Math. J. 53 (2001) 533-552.

[AK] S. Akbulut and R. Kirby, Branced covers of surfaces in 4-manifolds, Math. $Ann166$ (1966)

76-102.

[Ar] V.I.Arnold, Critical points of smooth functions, Proc. Intern. Congress Math. Vancouver (1974)

19-39.

[AGV] V.I.Arnold, S.M.Gusein-Zade A.N.Varchenko, Singularities of Differentiable Maps, Volume
Monographs in Mathematics. 83 (Birkhauser Boston, Inc., Boston, MA., 1988).

$\mathrm{I}\mathrm{I}$.

5Pacific J. Math.
[B]

K. Baker, (dissertation, in preparation.)

surger$ry$ on knots, Ann. Math. 125[CGLS] M. Culler, M. Gordon, J. Luecke and P. Shalen, Dehn surgery(1987) 237-300.
[D]

A. Dimca, Singularities and Topology of Hypersurfaces Universitext, Springer-Verlag, 1992.

[FS] R. Fintushel R. Stern, Constructing Lens spaces by surgery on knots, Math. Z. 175 (1980) 33-51.
[GI1] W. Gibson and M. Ishikawa, Links and gordian numbers associated with generic immersions ofintervals, Topology Appl. 123 N0.3 (2002) 609-636.
[GI2] W. Gibson and M. Ishikawa, Links of oriented divides and fibrations in link exteriors, OsikaOsakaMath. 39 N0.3 (2002) 681-703.

.

$J$

187
[GHY] H. Goda, M. Hirasawa Y. Yamada, Lissajous curves asfiber surfaces, Tokyo J. Math. 25 N0.2 (2002) 485-491.

$A$’Campo divides, torus knots and their[GZ] S. M. Gusein-Zade, Intersection matrices for certain singularities of functions of two valuables,
Functional Analysis and its Appl. 8 (1974) 10-13.

[HKK] J. Harer, A. Kas and R. Kirby, Handlebody decompositions of complex surfaces. $Mem$ . Amer.
Math. Soc. 62 (1986), no. 350,pp.
$\mathrm{i}\mathrm{v}+102$[H]

M. Hirasawa, Visualization ofits Appl. 121 (2002) 287-304.

[K]

R. Kirby, The topology of 4-manifolds, Lecture Notes in Math.,$A$’Campo’s fibered links and unknotting operations, Topology and$\mathrm{v}\mathrm{o}\mathrm{l}.1374$, Springer, 1989.

[KM] R. Kirby and P. Melvin, The E%-manifold, singular fibers and handlebody decompositions, PrOceedings of the Kirbyfest (Berkeley, $CA$ , 1998),-f258 (electronic), Geom. Topol. Monogr.,
Geom. Topol. Publ., Coventry, 1999.

$l$$\mathit{2}SS$[Mz] J. Matsuzawa, Arnold ’s strange duality to Casson- Walker Huhen-Ryo, (in Japanese) “Knots andLow Dimensional Topology” (1999), 26-35.
[Or] P. Orlik, Seifert manifolds, Lecture Notes in Math.,$\mathrm{v}\mathrm{o}\mathrm{l}.291$, Springer, 1972.

[Y1] Y. Yamada, Surgery along a projective plane in a 4-manifold and $D_{4}- sing^{qt}laity$. , Newton polyhedra and singularities (Japanese) (Kyoto, 2001) Surikaisekikenkyusho Kokyuroku No. 1233 (2001),102-110.

[Y2] Y. Yamada, Berge knots in the fiber surfaces of genus one, lens spaces and framed links, preprint(2003).
$\prime s$[Y3] Y. Yamada, Finite Dehn surgery along

YAMADA Yuichi$A$.ac . jp
Dept. of Systems Engineering
The Univ. of ElectrO-Communications
1-5-1, Chofugaoka, Chofu,
Tokyo, 182-8585, JAPANyyyamada(Dsugaku.

$\mathrm{e}$-one.

$\mathrm{u}\mathrm{e}\mathrm{c}$’Campo ’s divide knots , preprint (2004).