% \subsection{Overlays}
In the following, all skew Schur functions are considered as functions of the same set
of variables
$\pas{x_1,\dots,x_{\n}}$. (Equivalently, all tableaux have entries from the set
$\setof{1,\dots,\n}$, and all families of \nilp s have lower
points on level $1$ and upper
points on level $\n$).

By \eqref{eq:skewSchurNILP} %{eq:skewSchur} % and the \lgt\bindestrich interpretation of Young tableaux as \nilp s,
we may view the product of two skew Schur functions as the generating function of
pairs of $m$\bindestrich tuples of \nilp s
$$
\schurf_{\lambda/\mu}\cdot\schurf_{\sigma/\tau} =
	\sum_{\pas{P_1,P_2}}\weight\of{P_1}\cdot\weight\of{P_2}.
$$
Here, the sum runs over all pairs $\pas{P_1,P_2}$, where $P_1$ is a $\length{\lambda/\mu}$\bindestrich tuple
of \nilp s corresponding to the shape $\lambda/\mu$ and $P_2$ is a
$\length{\sigma/\tau}$\bindestrich tuple
of \nilp s corresponding to the shape $\sigma/\tau$.
Imagine that the lattice paths of $P_1$ and $P_2$ are coloured red and green, respectively:
This will give an \EM{overlay} of lattice paths,
%which also depends on (the difference of)
%the shift paramaters $t_1$ and $t_2$;
see \figref{fig:nilpOverlay} for an illustration.

\begin{figure}
\caption{Illustration: A green $7$\bindestrich tuple of \nilp s, corresponding to
shape $\lambda/\mu$, and a red $7$\bindestrich tuple of \nilp s, corresponding to
shape $\sigma/\tau$, constitute an \EM{overlay} of \nilp s. Here, $\sigma=\pas{7, 4, 4, 3, 1, 1, 1}$, $\tau=\pas{3, 2, 2, 1}$,
$\lambda=\sigma+\constpart{1}{7}$ and $\mu=\tau+\constpart{1}{7}$.
(Note that $\schurf_{\lambda/\mu} = \schurf_{\sigma/\tau}$.) The red and green paths are drawn with a slight offset for graphical
reasons, the colour red is indicated by dashed lines,
the colour green is indicated by dotted lines.
The picture also shows the Young tableaux corresponding to the red and green
$7$\bindestrich tuples of \nilp s.
}
\label{fig:nilpOverlay}
\begin{center}
\input graphics/nilpOverlay-slide
\end{center}
\end{figure}


Such overlays 
%``overlays of two families of nonintersecting lattice paths''
%as the generating function of ``overlays of two families of nonintersecting lattice paths''
%(according to definition~\eqref{eq:skewSchur})
give rise to a bijective construction,
which (to the best of our knowledge) was first used by Goulden \cite{goulden:1988}.
The same construction was used
in \cite{fulmek:2001} to describe a class of Schur function identities, special
cases of which amount to Dodgson's condensation formula and the Pl\"ucker relations.

%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}
%\caption{Illustration for the example given in Section~\ref{sec:bicoloured}.}
%\label{fig:twoSkewShapes}
%\input graphics/twoSkewShapes
%\end{figure}

We shall present this construction by way of an example: Consider skew shapes
$\lambda/\mu$ and $\sigma/\tau$, where $\sigma=\pas{7, 4, 4, 3, 1, 1, 1}$, $\tau=\pas{3, 2, 2, 1}$,
$\lambda=\sigma+\constpart{1}{7}$ and $\mu=\tau+\constpart{1}{7}$,
%For the skew shape $\lambda/\mu$, choose fixed shift $t_1=1$,
%and for the skew shape $\sigma/\tau$, choose fixed shift $t_2=0$. (See
see \figref{fig:nilpOverlay} for an illustration.

%$\lambda/\mu$, where
%\begin{align}
%\lambda &= \pas{14, 13, 13, 11, 11, 9, 9, 8, 8, 7, 5, 3},\notag\\
%\mu &= \pas{9, 9, 9, 6, 6, 5, 4, 4, 4, 3, 1},\label{eq:lambdamu}
%\end{align}
%and $\sigma/\tau$, where
%\begin{align}
%\sigma &= \pas{14, 14, 12, 12, 11, 11, 11, 9, 8, 7, 7, 5},\notag\\
%\tau &= \pas{10, 10, 8, 8, 8, 7, 6, 6, 6, 5, 2}.\label{eq:sigmatau}
%\end{align}
%For the skew shape $\lambda/\mu$, choose fixed shift $2$, and for the skew shape
%$\sigma/\tau$ choose fixed shift $0$, and consider the lower and upper points
%of the corresponding families of nonintersecting lattice paths. For instance,
%the upper point of the first path corresponding to $\lambda/\mu$ is $\pas{\lambda_1-1+2,N}=\pas{15,N}$, and the lower point of the last (twelfth) path corresponding to $\lambda/\mu$
%is $\pas{\mu_{12}-12+2,1}=\pas{-10,1}$, and the upper point of the first path corresponding
%to $\sigma/\tau$ is $\pas{\sigma_1-1,N}=\pas{13,N}$.


Now consider the arcs, lower points and upper points of some pair $\pas{P_1,P_2}$ of 
% families of
$7$\bindestrich tuples of lattice paths,
where $P_1$ corresponds to shape $\lambda/\mu$ and $P_2$ corresponds to
shape  $\sigma/\tau$.
%Such arcs/points are coloured 
%\bit
%\item \EM{green} if they correspond to $\lambda/\mu$,
%\item \EM{red} if they correspond to $\sigma/\tau$.
%\eit
Note that there may be arcs/points coloured both green \EM{and} red. Such arcs/points will
never be affected by the following constructions: We call them \EM{uncoloured}; the \EM{remaining} arcs/points
% such arcs and points the \EM{uncoloured}
%points (``colours green and red cancel each other''), the lower and upper points
(which are coloured \EM{either} red \EM{or}
green) are called the \EM{coloured} arcs/points.

%Note that the number of coloured points is necessarily
%\EM{even}.
%
%
%Now consider some pair $\pas{P_1,P_2}$ of families of nonintersecting lattice paths,
%where $P_1$ corresponds to shape $\lambda/\mu$, and $P_2$ corresponds to
%shape  $\sigma/\tau$.
%We call the paths of $P_1$ the \EM{green} paths and the paths of $P_2$ the \EM{red} paths,
%and we colour the arcs of the lattice $\mathbb Z^2$ accordingly (i.e., arcs used by some
%white path are coloured white, and arcs used by some black path are coloured black).
%As with the lower/upper points, arcs which are coloured black \EM{and} white are
%not affected by the following construction, and we call all arcs which are
%\EM{either} black \EM{or} white the \EM{coloured} arcs.
We construct \EM{bicoloured trails}
\bit
\item connecting coloured points
\item and using (only) coloured arcs
\eit
by the following algorithm: 
% \begin{quote}

\begingroup
\leftskip1em
\rightskip1em
\em
We start at some coloured point $s$ and identify the
\EM{unique} coloured arc $\arca$ incident with $s$ which is \EM{of the same colour} as $s$.
Then we follow the lattice path starting in $\arca$ in the implied direction (i.e., either
up/right if
$s$ is a lower point, or down/left if $s$ is an upper point).

Whenever we meet \EM{another} path on our way (necessarily, this path is of the other colour),
we ``change colour and direction'', i.e., we follow this new path \EM{and} change the direction (if we were moving up/right along the old path, we move down/left
along the new path, and vice versa). Note that such change of colour and orientation might
also occur at the very beginning: For instance, if $s$ is a green upper point, but there
are (precisely) two red arcs incident with $s$, then we follow the red arc in the right direction. 

We stop if there is no possibility to go further,
i.e., if we end in another coloured point.

\endgroup
% \end{quote}

%For the coloured points, assume the circular orientation ``from right to left along level $N$,
%and then from left to right along level $1$''.
%In the upper picture of \figref{fig:twoSkewShapes}, this circular orientation is indicated by a grey circular
%arrow.

%Furthermore, assign to paths corresponding to
%% some tableau in the expansion (according to \eqref{eq:skewSchur}) of
%$\lambda/\mu$ the orientation \EM{downwards}, and to paths corresponding to
%% some tableau in the expansion of
%$\sigma/\tau$ the orientation \EM{upwards}. In the upper picture of \figref{fig:twoSkewShapes}, this orientation of paths is indicated by upwards or downwards pointing triangles.

\figref{fig:bicolouredPaths-slide} illustrates this construction (see also \cite{fulmek:2001}).

\begin{figure}[h]
\caption{%Recolouring bicoloured paths.
The picture shows the bicoloured trails (as thick grey lines with rounded corners)
for the example from \figref{fig:nilpOverlay}.}
\label{fig:bicolouredPaths-slide}
\begin{center}
\input graphics/bicolouredPaths-slide
\end{center}
\end{figure}

The following observations are immediate from the construction:
\begin{obs}[{\bf Bicoloured trails always exist}]
\label{obs:arbitrary-point}
For \EM{every} coloured point $s$, there \EM{exists} a bicoloured trail starting at $s$.
\end{obs}

\begin{obs}[{\bf Bicoloured trails can never cross}]
\label{obs:different-parity}
Bicoloured trails may have lattice points in common (they may intersect),
but they can never \EM{cross} (see \figref{fig:noncrossing}).
\end{obs}

\begin{figure}[h]
\caption{Bicoloured trails can never cross: Since bicoloured
trails never use arcs coloured both red \EM{and} green, a meeting point of bicoloured trails
can (up to interchanging colours red and green) only occur in the situations shown below. By 
construction, the bicoloured trails necessarily run as indicated by the grey rounded lines:
They do not cross.}
\label{fig:noncrossing}
\begin{center}
\input graphics/noncrossing
\end{center}
\end{figure}


Now consider some bicoloured trail $b$ in the overlay of \nilp s $\pas{P_1,P_2}$: 
Changing colours (green to red and vice versa)
\bit
\item of both ending points of $b$
\item and of all arcs of $b$
\eit
gives a new overlay of \nilp s  $\pas{P_1^\prime,P_2^\prime}$
(with different lower/upper points, see \figref{fig:recolouringPaths-slide}).
Clearly, we have:
% It is obvious that we have for this \EM{recolouring} of a bicoloured path:
\begin{obs}[{\bf Recolouring bicoloured trails is a weight preserving involution}]
\label{obs:weight-preserving-involution}
The recolouring of a bicoloured trail $b$ in an overlay  of \nilp s $\pas{P_1,P_2}$ is an \EM{involutive operation}, i.e., if we obtain
the overlay $\pas{P_1^\prime,P_2^\prime}$ by recolouring $b$ in $\pas{P_1,P_2}$, then
recolouring $b$ \EM{again} in $\pas{P_1^\prime,P_2^\prime}$ yields the original
$\pas{P_1,P_2}$. Moreover, this operation \EM{preserves % (the product of)
the corresponding weights}, i.e.,
$$
\weight\of{P_1}\cdot\weight\of{P_2}=\weight\of{P_1^\prime}\cdot\weight\of{P_2^\prime}.
$$
\end{obs}

\begin{figure}[h]
\caption{Illustration. Recolouring two bicoloured trails (indicated by thick grey
lines) from \figref{fig:bicolouredPaths-slide} changes the colour of lower and/or
upper points, thus changing the corresponding shape (shown below the paths).}
\label{fig:recolouringPaths-slide}
\begin{center}
\input graphics/recolouringPaths-slide
\end{center}
\end{figure}

Note that the operation of recolouring bicoloured trails changes the colours (red/green)
of coloured lower and/or upper points (which implies a change of the corresponding shapes,
see \figref{fig:recolouringPaths-slide}).
%so it is not immediately obvious on which set
%this involution actually is defined.
We want to encode this change in a convenient way: Imagine that the
coloured points are
arranged on a circle (see \figref{fig:bicolouredPaths2-slide}). Assign to point $s$
the \EM{radial orientation} (with respect to this circle)
\bit
\item \EM{inwards}, if $s$ is a \EM{red upper point} or a \EM{green lower point},
\item \EM{outwards}, if $s$ is a \EM{green upper point} or a \EM{red lower point}.
\eit

%\begin{tabular}{|c|c|c|}
%      & red      & green \\
%upper & inwards  & outwards \\
%lower & outwards & inwards
%\end{tabular}

\begin{figure}[h]
\caption{
The ``radial orientation'' (indicated in the picture by inward/outward pointing triangles) of
coloured points (shown as white circles) in their arrangement on a circle (shown as grey rectangle with rounded corners) depends on their position (upper or lower) and colour (red or green).
}
\label{fig:bicolouredPaths2-slide}
\begin{center}
\input graphics/bicolouredPaths2-slide
\end{center}
\end{figure}

See \figref{fig:bicolouredPaths2-slide}. From the construction of bicoloured trails,
the following is immediate:
\begin{obs}[{\bf Bicoloured trails connect points of different radial orientation}]
\label{obs:different-orientation}
Bicoloured trails never connect points of the same radial orientation
(i.e., two points oriented both inwards or both outwards).
\end{obs}

Clearly, we may ``forget'' the \EM{colours} (red or green) of the coloured points,
if we remember instead their \EM{radial orientations}: The situation is completely
determined by
\bit
\item the geometric positions of the (coloured and uncoloured) lower and upper points,
	we call this piece of information the \pointconf,
\item together with the radial orientation of the coloured points; we call this piece of information
	the \pointarr.
\eit
\figref{fig:simplifiedPicture} illustrates this.

\begin{figure}[h]
\caption{%Coloured points may be green or red:
The pattern of radial
orientations \pointarr\ (indicated by inward/outward pointing triangles)
encodes the actual colours (red or green) of the coloured points in the \pointconf\ 
(indicated by light grey horizontal bars in the picture; coloured/uncoloured points are shown as
white/grey circles there).
% and thus determines the corresponding shapes $\lambda/\mu$ and $\sigma/\tau$.
The bicoloured trails constitute a
\EM{noncrossing perfect oriented matching} (\nop).
The picture illustrates this for the example from
\figref{fig:bicolouredPaths2-slide} (the edges of the \nop\ are indicated by grey arcs).}
\label{fig:simplifiedPicture}
\begin{center}
\input graphics/simplifiedPicture
\end{center}
\end{figure}

Note that for all \pointconf s, the number of coloured \EM{upper} points plus
twice the number of uncoloured \EM{upper} points equals the number of coloured
\EM{lower} points plus twice the number of uncoloured \EM{lower} points.

We call a \pointarr\ \EM{admissible} if
it has the same number of inwardly/outwardly
oriented points.
\EM{Every} admissible \pointarr\ determines together with the
corresponding \pointconf\ a certain colouring (green and red) of lower/upper points,
which represents a certain product of two skew Schur functions
$${\schurf_{\lambda/\mu}}\cdot{\schurf_{\sigma/\tau}},$$
where
\bit
\item $\lambda/\mu$ is determined by the uncoloured points and the green points,
\item $\sigma/\tau$ is determined by the the uncoloured points and the red points.
\eit
Note that there might be no overlay of families of \nilp s that connect
these points (if, for instance, the $i$\bindestrich th green upper point lies to the left of the
$i$\bindestrich th green lower point; this would correspond to an $i$\bindestrich th row of length $<0$
in the corresponding Ferrers diagram): In this case, the corresponding skew Schur function
${\schurf_{\lambda/\mu}}$ is
zero.

But if there \EM{is} an overlay $\pas{P_1,P_2}$ of families of \nilp s that connect
the green and red points, then we may
imagine that the \pointarr\ corresponding to this overlay %of nonintersecting lattice paths
is arranged on a perfect (strictly convex) circle:
Observations~\ref{obs:arbitrary-point}, \ref{obs:different-parity} and
\ref{obs:different-orientation} imply that if we draw a straight line connecting two points in the \pointarr\ whenever the corresponding coloured points
are connected by a bicoloured trail in $\pas{P_1,P_2}$, then we will obtain a
\pointmatch\
% \EM{perfect noncrossing oriented matching}
$\cnr{m}$, i.e.:
%Recall that we have by Observations~\ref{obs:different-parity} and \ref{obs:different-orientation}:
\bit
\item \EM{Every} coloured point is incident with a straight line (edge) in $\cnr{m}$,
\item Every edge in $\cnr{m}$ connects points of \EM{different radial orientation},
\item No two edges in $\cnr{m}$ do \EM{cross} (in the geometric realization as straight lines
	connecting points on a circle).
\eit
% We call a matching $M$ in a \pointarr\ \EM{admissible} if it satisfies the above conditions.

In particular, this implies:
\begin{obs}[{\bf Bicoloured trails span equal numbers of inward and outward directed points}]
\label{obs:equal-sizes}
Assume $p$ and $q$ are coloured points in an overlay of \nilp s which are connected by some bicoloured trail. Then the corresponding
\pointarr\ is divided in two parts by the points corresponding to $p$ and $q$:
%(the circle is cut in two halves by removing $p$ and $q$): 
In each of these
parts, the number of points directed inwards must equal the number of points directed outwards.

Stated otherwise:
Let $\cnr{m}$ be a \pointmatch\ for a \pointarr\ $\cnr{r}$. An arbitrary edge $e$ in $\cnr{m}$ divides 
$\cnr{r}$ into two parts, each of which must contain equal numbers of inward and outward oriented
points.
\end{obs}

%We may summarize
%then the family of all bicoloured trails determines a \EM{perfect matching} $M$
%in the \pointarr\ (according to Observation~\ref{obs:arbitrary-point}), which is \EM{non\bindestrich crossing}
%(according to Observation~\ref{obs:different-parity}), and where all edges of $M$
%connect points of different radial orientation (according to
%Observation~\ref{obs:different-orientation}): We call such matchings \EM{admissible}. Note that recolouring some bicoloured
%trail amounts to reversing the orientation of the corresponding edge in the matching $M$.
%%
%
%If we focus on the coloured points, we may encode the situation in a simpler
%picture, where the coloured lower/upper points are located on the lower/upper half of a \EM{circle}, and where the orientation of
%the respective path is translated to a \EM{radial orientation} (either towards the
%center of the circle or away from it). The lower picture of \figref{fig:twoSkewShapes}
%illustrates this: A grey horizontal line indicates the separation of the lower and upper half
%of the circle, the point labeled $1$ corresponds to the lattice point $\pas{15,N}$,
%the point labeled $2$ corresponds to the lattice point $\pas{6,N}$, and so on.
%
%Now consider some pair $\pas{P_1,P_2}$ of families of nonintersecting lattice paths,
%where $P_1$ corresponds to some tableau of shape $\lambda/\mu$, and $P_2$ corresponds to
%some tableau of shape  $\sigma/\tau$.
%We call the paths of $P_1$ the \EM{white} paths and the paths of $P_2$ the \EM{black} paths,
%and we colour the arcs of the lattice $\mathbb Z^2$ accordingly (i.e., arcs used by some
%white path are coloured white, and arcs used by some black path are coloured black).
%As with the lower/upper points, arcs which are coloured black \EM{and} white are
%not affected by the following construction, and we call all arcs which are
%\EM{either} black \EM{or} white the \EM{coloured} arcs. We 
%construct \EM{bicoloured paths}
%\bit
%\item connecting (only) coloured lower/upper points
%\item and using (only) coloured arcs
%\eit
%by the following algorithm: 
%\begin{quote}
%We start at some coloured point $q$ and follow
%the path determined by the \EM{unique} coloured arc incident with it in the respective orientation (i.e., either up/right or down/left). Whenever we meet \EM{another} path on our way (necessarily, this path is of the other colour),
%we ``change colour and orientation'', i.e., we follow this new path \EM{and} change the orientation (i.e., if we were moving up/right along the old path, we move down/left
%along the new path, and vice versa). We stop if there is no possibility to go further.
%\end{quote}
%
%\begin{obs}[{\bf Bicoloured path connect points of different parity}]
%\label{obs:different-parity}
%Two different bicoloured paths may have lattice points in common (they may intersect),
%but they can never \EM{cross}.
%%
%If we assume some consecutive numbering of the coloured points in their circular
%orientation (see the lower picture of \figref{fig:twoSkewShapes}), then this non\bindestrich crossing condition implies that there can never be a bicoloured path connecting two points with numbers of the same parity.
%\end{obs}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}
%\caption{%Recolouring bicoloured paths.
%The left picture presents two tableaux of the identical shape
%$\pas{7, 4, 4, 3, 1, 1, 1}/\pas{3, 2, 2, 1}$ in the lower part, the upper part shows the
%corresponding overlay of black and white paths (white paths are indicated by dashed lines)
%and \EM{all} bicoloured paths (indicated by thick grey lines). The
%right picture presents two tableaux of shapes $\pas{5, 3, 3, 2, 2, 1, 1, 1}/\pas{2, 1, 1, 1, 1}$
%and $\pas{9, 5, 5, 1, 1, 1}/\pas{4, 3, 1}$, respectively, in the lower part: These tableaux
%correspond to the overlay of lattice paths in the upper part, which is obtained by
%\EM{recolouring} the bicoloured paths starting in $\pas{7,8}$ and in $\pas{-1,1}$ (these
%bicoloured paths are indicated by thick grey lines, again.}
%\label{fig:bicolouredPaths}
%\input graphics/bicolouredPaths
%\\
%\input graphics/bicolouredPaths-slide
%\\
%\input graphics/recolouringPaths-slide
%\end{figure}
%
%\begin{figure}
%\caption{Illustration.}
%\input graphics/bicolouredPaths2-slide
%\end{figure}

%\begin{figure}
%\caption{Illustration \gedankenstrich\ was?.}
%\input graphics/GenericConfiguration-slide
%\end{figure}
%

We conclude our preparatory considerations with the following observation:
\begin{obs}[{\bf Admissible matchings always can be realized by overlays of \nilp s}]
\label{obs:realizematching}
For every
% pair $\pas{\cnr{c},\cnr{r}}$, where $\cnr{c}$ is a \pointconf\ and $\cnr{r}$
% is an admissible \pointarr\ for $\cnr{c}$, 
admissible \pointarr\ $\cnr{r}$,
there exists a % pair of shapes
\pointconf\ $\cnr{c}$
such that \EM{every} \pointmatch\ % $\cnr{m}$
in $\cnr{r}$ can be \EM{realized} by an overlay of \nilp s corresponding to % the pair of shapes $\pas{\lambda/\mu,\sigma/\tau}$ corresponding to
$\pas{\cnr{c},\cnr{r}}$:
%$\pas{\lambda/\mu,\sigma/\tau}$ characterized by $\pas{\cnr{c}^\prime,\cnr{r}}$, which has
%the \EM{same} upper and lower uncoloured, green and red points as $\pas{\cnr{c},\cnr{r}}$
%in the \EM{same} order (i.e., only the \EM{geometric position} of the points may differ),
%such that \EM{every} \pointmatch\ $\cnr{m}$ in $\cnr{r}$ can be \EM{realized} by an overlay of \nilp s corresponding to $\pas{\lambda/\mu,\sigma/\tau}$:
Maybe the best way to conceive this is by looking at pictures, 
see \figref{fig:RealizeMatching} and \figref{fig:RealizeMatching2}.
\end{obs}

\begin{figure}[!h]
\caption{Realizations of matchings, 1st part. Assume we want to realize a \EM{partial}
\pointmatch\ for some subset $S$ of \EM{coloured} upper points which are \EM{consecutive}
in the corresponding \pointarr. In the corresponding
\pointconf, there might be uncoloured points which are interspersed between the coloured
points. As an example, consider the
partial \pointmatch\ indicated in the upper picture.
%Uncoloured points are indicated by
%black circles.
Imagine that the red and green paths are
threads dangling down from the corresponding points: Note that \EM{two} threads are dangling
down from \EM{uncoloured points} (shown as black circles), while only \EM{one} thread is
dangling down from \EM{coloured points} (shown as white circles). Repeat the following step until there
are no more coloured points: 
%\EM{Forget} all the
%uncoloured points (there is only one in the upper picture, indicated by a grey circle)
%and
If two coloured points $p_1$, $p_2$
%\bit
%\item
are connected by an edge $e$ of the partial \pointmatch,
%\item
and all points between $p_1$ and $p_2$ (if any) are \EM{uncoloured},
%\eit
then arrange the threads dangling down from the points $p_1,\dots,p_2$ such that they
alternate in colour and tie together with knots the first and second, the third and fourth, etc.,
of these threads. View these knots as the new uncoloured points and simply \EM{forget}
the points $p_1,\dots,p_2$: Forgotten points are indicated by grey circles. The middle
picture shows an intermediate state, where three edges of the partial \pointmatch\ are
realized by bicoloured trails (shown as thick grey lines). The lower picture shows the final state, where
there are no more coloured points, and all edges of the partial \pointmatch\ are \EM{realized}
by bicoloured trails.
%all pairs of consecutive red/green threads dangling down from \EM{neighbouring coloured} points
%(i.e., with no coloured point between them) which are connected
%by an edge $e$ of the partial matching: Note that it is always possible to We call this operation the \EM{realization} of the
%edge $e$; the middle picture shows this for our example.
%Now we \EM{uncolour} all points incident with edges already realized and repeat the step just
%described until there are no more coloured points., then among the remaining
%points we again identify the neighbouring ones which are connected by edges of the partial
%matching, and fasten with knots all dangling threads
%between them (note that  the dangling threads can be brought
%into the order needed):
% The lower picture shows this for our example. 
It is clear that the
same algorithm realizes partial \pointmatch s
for subsets of \EM{consecutive coloured} lower points.}
\vspace*{-2ex}
\label{fig:RealizeMatching}
\begin{center}
\input graphics/RealizeMatching
\end{center}
\vspace*{-2ex}
\end{figure}
\newpage
\begin{figure}[!h]
\caption{Realizations of matchings, 2nd part. Assume that all maximal partial
\pointmatch s on subsets of \EM{consecutive coloured} lower or \EM{consecutive coloured} upper
points already have been
realized, so now we have to realize edges connecting a lower point to an upper point
where there is no coloured point between them (in the right half of the circular arrangement
of points).
The left picture indicates this situation: The edges already realized are shown as black
lines. As in \figref{fig:RealizeMatching}, coloured points are shown as white circles,
uncoloured points as black circles and forgotten points as grey circles.
%% Referee Punkt 6
Obviously, by ``tying together'' threads of different colours
(as in \figref{fig:RealizeMatching}) combined with ``weaving together'' upwards/downwards dangling threads of the same
%% Referee Punkt 6
colour, we can realize the edge (shown as a grey arc) that connects these
two coloured points. It is clear that
this algorithm indeed gives ``topological'' bicoloured trails corresponding to the
\pointmatch, and it is also clear, that the ``topological'' situation can be implemented
``geometrically'' with lattice paths for appropriately chosen shapes (with large enough distances between
the upper points and lower points).}
\label{fig:RealizeMatching2}
\begin{center}
\input graphics/RealizeMatching2
\end{center}
\vspace*{-4ex}
\end{figure}
%\clearpage

