Positive tree-like mapping classes

In this paper we will study special mapping classes of orientable
surfaces with one boundary component. A mapping class is an element
of the mapping class group. The mapping class group for an orientable
surface S with boundary is defined as
MCG(S) = Diff+(S; @S)=Diff+
0 (S; @S):
So a mapping class is an isotopy class of diffeomorphisms fixing the
boundary pointwise. Tree-like mapping classes are represented by dif-
feomorphisms, that are a product of Dehn twists along a system of
essential simple closed curves on the surface. Curves of the system
intersect at most once with another curve and the complement of the
system of curves is a cylindrical neighborhood of the boundary. We can
build a graph by representing each twist curve by a vertex and connect-
ing two vertices by an edge if the two corresponding curves intersect.
The mapping class is called tree-like, if this graph is a tree. When we
keep the information of cyclic ordering of the curves on the surface, we
get a planar tree, and call it the geometrical Dynkin diagram.
A tree-like mapping class is called positive, if all Dehn twists which
are performed are right or positive Dehn twists.
We will establish an algorithm to distinguish positive tree-like map-
ping classes up to conjugacy. The conjugacy problem for surface map-
ping classes has already been solved by Thurston, but in concrete ex-
amples it can be very hard to determine whether two mapping classes
are conjugate or not.
In the following we will speak of surface diffeomorphisms meaning
surface diffeomorphisms up to isotopy and mapping classes, respec-
tively. So we will define a diffeomorphism and regard it as a represen-
tative for a mapping class.
Positive tree-like diffeomorphisms arise as monodromies of a special
class of fibred knots, the slalom knots. These slalom knots can be con-
structed out of a rooted planar tree, which is related to the geometrical
Dynkin diagram.
Up to the exceptions E6, E8 and the series A2n the monodromies
of slalom knots are pseudo-Anosov. By a theorem of Thurston, there
exist two transverse measured foliations that are invariant under this
diffeomorphism. In chapter 3 we will give an explicit description of the
measured foliations for slalom knots with pseudo-Anosov monodromies.
The rooted planar tree will play again a crucial role. Measured foli-
ations have been studied under different aspects. Casson and Bleiler
considered in [BC] geodesic lamination and Strebel studied in [S] qua-
dratic differentials. For further studies see [FPL].
The two measured foliations of a monodromy are an invariant of the
diffeomorphism up to conjugacy. But since measured foliations contain
a lot of information that is not easy to take care of, it is very hard to
use it as a tool to distinguish concrete diffeomorphisms.
Particularly, it is very hard to distinguish slalom monodromies that
arise from the same abstract rooted tree, but from different embeddings
into the plane since the corresponding slalom knots are mutant. The
notion of mutation was introduced by Conway in [Co]. Mutant knots
are hard to distinguish. For small examples the quantum invariant can
be calculated and separates. Knotscape too, helps us to separate small
examples. Sometimes there is also a symmetry argument that can be
applied. But for the whole class of slalom knots it was not known if all
knots coming from different rooted planar trees were different.
In chapter 5 we give a solution to this problem. We introduce a
method to reconstruct the rooted planar tree out of the diffeomor-
phism by a geometrical algorithm for all diffeomorphisms that arise
from rooted planar trees with at least three crown vertices. So the
rooted planar tree is an invariant for the slalom knot, and hence all
slalom knots are different. Slalom knots with one or two crown ver-
tices arise from trees with only one planar embedding. The theory of
the Montesinos links can be applied to them, and separates them (see
[Tu]). Therefore we obtain the result, that all slalom knots coming
from non-congruent planar trees are different.
To get this algorithm, we need an important property of the slalom
monodromy. All slalom monodromies are strongly inversive. This
means, that there exists an involution, that conjugates the monodromy
to its inverse. This property is inherited from slalom knot. Slalom
knots are strongly invertible, so there exists an involution of S3 sending
the oriented knot to itself, fixing two points on the knot, and reversing
the orientation of the knot. If the knot is fibred, this involution can
be chosen to respect the fibers, and therefore the monodromy becomes
strongly inversive [To].
In chapter 4 we analyse these involutions. We will see, that up
to conjugacy of the pair (monodromy, involution), there are at most
two such involutions. Each of these involutions fixes an arc on the
surface pointwise. We will study these fixed arcs and their images under
the monodromy, and we will see, that the number of intersections of
the fixed arc and its image under the monodromy differ for the two
arcs coming from the two involutions. So the two involutions can be
distinguished using their fixed arcs. Furthermore, one of this fixed arcs
will play a crucial role in the reconstruction of the rooted planar tree.