**Norbert A'Campo**

Vortrag Arbeitstagung 1999 Bonn.

A divide is a relative generic immersion of a
1-manifold
in the unit 2-disk
.
In this talk a
divide will be an immersion of a finite union of copies of
[0,1] and will be called a picture.
To a picture
is associated a classical link in the 3-sphere.
Consider the tangent space
as
.
The link *L*(*P*) of the picture *P* is the following subset in the
unit sphere *S*^{3} of :

The link

Figure 1: A picture with 2 branches

The unknotting number of the link of a picture is computed by:

The proof relies on the positive answer of Kronheimer and Mrowka to the Thom conjecture.

For given natural numbers
there are infinitely many links
with *r* components and with unknotting number ,
but only a
finite number of transversal isotopy classes of pictures with *r*
branches and
double points. So we see that links *L*(*P*) of
pictures are as rare among links as orchids among flowers. It is a
surprise that the knots
10_{139}, 10_{145}, 10_{152} of the table
of Rolfsen, for which the
unknotting number has been determined only recently by Tomomi Kawamura [K],
are knots of pictures. The knot 10_{139} is very rare since it is a knot
of two
pictures
which can not be related by an transversal isotopy (see Fig. 2).

Figure 2: Two pictures for the knot 10

Links of pictures tend to be fibered.

The proof uses a complex function *f*_{P} on
which satisfies
the Cauchy-Riemann equations along the zero section *D* in *TD*, such that
the 0-level of *f*_{P} intersects *D* along *P*.

The geometric monodromy can be read from the picture in a similar way as the monodromy of a plane curve singularity from a picture provided by a small local real deformation of the singularity with the maximal possible number of double points in .

Let the polynomial function
has at
an
isolated singularity. We may assume without restricting the
local topology of the singularity that
the factorization of *f* into iReducible local
branches has only real factors. Let
be the local link of the singularity of *f* at 0.
Let
be the picture of some small real deformation
with the maximal number
of local double points
of the singularity. Hence, the Milnor number of the singularity is
,
where
*r* is the number of local branches (see AT1974).
The following theorem gives new insight for the local topology
of plane curve singularities.

An embedded tree *B* in the unit disk *D*, such that the
intersection
consists of one
terminal vertex *r* of *B*, is called a
rooted planar tree. For a rooted planar tree *B* there exists an immersed copy
of the
interval [0,1] with the following properties:

(i) The immersion is relative, i.e. the endpoints are embedded in .

(ii) The immersion is generic, i.e. there are only transversal crossing points, only the endpoints lie on and the immersion is transversal to .

(iii) The double points of *P*_{B} lie in the interior of the edges of
*B*, such that the local branches are transversal to the edge of *B*.

(iv) Each connected component of
contains
exactly one vertex of
*B*.

(v) The only intersection points of *P*_{B}
with *B* are the double points of *P*_{B}.

The immersed curve *P*_{B} is well-defined up to
regular relative isotopy
and is called the slalom curve of the rooted
planar tree *B*, see Fig. 3. The left picture of Fig. 2 is a slalom curve.

Figure 3: Rooted planar tree, its Dynkin diagram

The Dynkin diagram
of the
slalom curve *P*_{B} is deduced from the rooted tree *B* as follows:
First make a new tree *B*' by subdividing each edge of *B* with a new
vertex, which is placed at the crossing point of *P*_{B} on the edge; next,
remove from *B*' the root vertex *r* and the terminal edge of
*B*'
pointing to *r*. In Fig. 3 the tree *B* has the shape of the classical
Dynkin diagram *D*_{6} but the Dynkin diagram
of *P*_{B} has 10
vertices and we can denote it by *E*_{10}.
The Dynkin diagram
of a rooted tree *B* is a bicolored rooted
tree with an embedding in the plane.
The root is the new vertex which lies on the edge of
*B* originating from the root
point of *B* and the bicoloring is such that the new
vertices are of the same color. Moreover, the Dynkin
diagram
has the
property that the terminal vertices of
different
from the root, are never
new.

Using in addition to rooted trees also ``disk-wide-webs'' the theorem has a formulation which includes also extended Dynkin diagrams.

If the Dynkin diagram
is among
,
the knot
*K*_{B} is the torus knot (2,2*k*+1), (3,4) or (3,5) and appears
as the local knot of a simple
plane curve singularity; the monodromy diffeomorphism (with free boundary)
of the knot *K*_{B} can be chosen to be
of finite order in those cases and its complement does not carry a
complete hyperbolic metric.

Figure 4: Fundamental domain for the complement of the slalom knot

With the help of SnapPea and Snap, one gets that the
complement *M* of the slalom knot *E*_{10}
with its hyperbolic structure is
triangulated by 3 isometric ideal simplices with cross ratios
,
where *z* is the root with negative imaginairy part of *x*^{3}-*x*^{2}+1=0
(see Fig. 3, ).
Hence
*B*=-*z*^{2} + *z*, *A*=-2/5*z*^{2}+3/5*z*+1/5, *C*=-*z*^{2}+*z*-1. The
fundamental group of *M* is generated by 2 elements, which can be chosen
such that the coResponding hyperbolic motions are the fractional
transformations

Figure 5: The slalom knot

From the above theorem we get many examples of hyperbolic fibered knots, whose monodromy diffeomorphism and gordian number are known explicitly. The monodromy diffeomorphism of a slalom knot can be realized as a product of right Dehn twists of a system of simple closed curves on the fiber surface, such that the union of the curves is a spline in the fiber surface and the dual graph of the system is the Dynkin diagram of the rooted tree; the gordian number of a slalom knot equals the number of crossings of the slalom divide. We call the isotopy class of the monodromy diffeomorphism of the slalom knot of a rooted tree the Coxeter diffeomorphism of the Dynkin diagram of the rooted tree. It follows from the theorem and a theorem of Thurston that a Coxeter diffeomorphism of the Dynkin diagram of a rooted tree is pseudo-anosov if and only if the Dynkin diagram is not a classical Dynkin diagram.

Slalom knots can be characterized by the following properties:

- the knot is prime and fibered,

- the monodromy has minimal complexity 4*g*-1, where *g* is the genus
of the knot,

- the monodromy is a product of Dehn twists, which all belong to the same conjugacy class in the mapping class group.

The complexity of an element *T* of the mapping class group, which we use here,
is the minimum of the quantity *a*+*b* over all the
product decompositions of *T* as product of Dehn twists, where *a* is the
length of the product and where *b* is the number of mutual
intersection points of
the core curves of the twists involved in the product decomposition.

[AC1]
Norbert A'Campo,
*Real deformations and complex topology of plane curve singularities*,
Annales de la Faculté des Sciences de Toulouse,
(1999), to appear.

[AC2]
Norbert A'Campo,
*Generic immersions of curves, knots,
monodromy and gordian number*,
Publ. Math. IHES, to appear.

[AC3]
Norbert A'Campo,
*Planar trees, slalom curves and hyperbolic knots*,
Publ. Math. IHES, to appear.

[K]
Tomomi Kawamura,
*The unknotting numbers of 10 _{139} and 10_{152} are 4*,
Osaka J. Math.