[TorusAlgebra: some improvements, joint with Peter
lipshitz@math.columbia.edu**20120102160654
 Ignore-this: f9216182c14606df47a6cbe3e168a8ca
] hunk ./TorusAlgebra.tex 222
-  procedure.  The procedure produces a collection of tiles $T$,
-  arranged in a rooted tree which we call the {\em gluing tree}, and
-  which are glued along arcs (hence the
-  result is diffeomorphic to the standard disk). It also specifies a
-  assignment from arcs in the boundary instruction to tiles.
-  
+  procedure.  The procedure produces:
+  \begin{itemize}
+  \item A collection of tiles $T$, arranged in a rooted tree which we
+    call the {\em gluing tree}.
+  \item A surface obtained by gluing tiles along arcs as specified by
+    the gluing tree.  (This surface is homeomorphic to a disk.)
+  \item An assignment from arcs in the boundary instruction to tiles
+    in the gluing tree.
+  \end{itemize}
hunk ./TorusAlgebra.tex 236
-  arc $a_{i+1}$ in the boundary instruction. There are three  cases.
+  arc $a_{i+1}$ in the boundary instruction. There are three cases.
hunk ./TorusAlgebra.tex 249
-  but $a_i$ {\em is} the fourth last arc on the boundary of $T$ encountered so far, then
-  $a_{i+1}$ is associated to the predecessor in the gluing tree of $T$. %
-  \smargin{RL: what does ``$a_{i+1}$ is associated to the predecessor
-    to $T$'' mean?}%
-  \smargin{Clearer now?}
+  but $a_i$ {\em is} the fourth arc on the boundary of $T$ encountered
+  so far, then
+  $a_{i+1}$ is associated to the predecessor in the gluing tree of
+  $T$. (This case does not involve any new gluing.)
hunk ./TorusAlgebra.tex 256
-  tile).
+  tile). (This case does not involve any new gluing.)
hunk ./TorusAlgebra.tex 260
+\smargin{RL requests a figure, and a synopsis of which cases occur at
+  each step in the figure.}
+
hunk ./TorusAlgebra.tex 264
-which is tiled by copies of $T_0$, one of which is initial. %
-\smargin{RL: I don't understand what the surface is in the
-  non-admissible case.}%
-\smargin{PSO: let's discuss this}
+which is tiled by copies of $T_0$, one of which is initial.
hunk ./TorusAlgebra.tex 272
-    admissible}
-  if, when we apply Construction~\ref{constr:BoundaryInstructions},
-  the resulting tiles satisfy the following two properties:
+    admissible} if, when we apply
+  Construction~\ref{constr:BoundaryInstructions}, the resulting tiles
+  satisfy the following two properties:
hunk ./TorusAlgebra.tex 276
-    \item \label{req:FourArcsPerTile} For each tile $T$, there are four arcs assigned to $T$
-      under the above construction;
-    \item \label{req:OrderOfArcs} The four arcs corresponding to any
-      given
-      tile $T$ appear in the
-      cyclic
-      order specified by $T_0$.
+  \item \label{req:FourArcsPerTile} For each tile $T$, there are four
+    arcs assigned to $T$ under the above construction;
+  \item \label{req:OrderOfArcs} The four arcs assigned to any given
+    tile $T$ appear in the cyclic order specified by $T_0$.
hunk ./TorusAlgebra.tex 283
+\smargin{RL requests some examples of admissible versus non-admissible
+  boundary instructions}
+
hunk ./TorusAlgebra.tex 435
-  Applying Construction~\ref{constr:BoundaryInstructions},
-  to a left-extended sequence, we find that the first $i$ arcs, say,
-  are associated to tiles, and each of those tiles is assigned only
-  one arc. Thus, the sequence is clearly not admissible. Moreover,
-  if we factor  $a_1\uplus\dots \uplus a_i$ of $r_1$, we obtain an
-  admissible sequence. Similarly, for a right-extended sequence,
-  if $a_m$ is the last arc in the initial tile, then the sequence is
-  right-extended by $a_{m+1}\uplus\dots\uplus a_n$.
-  \smargin{RL didn't follow this proof.}
+  Consider first a left-extended sequence, in which $r_1'$ is the
+  juxtaposition of $i$ arcs.  Applying
+  Construction~\ref{constr:BoundaryInstructions}, the first $i$ arcs
+  are associated to different tiles, and each of those tiles is
+  assigned only one arc. Thus, the sequence is not strongly admissible
+  or right-extended admissible. Further, to obtain an admissible
+  sequence we must factor at least $i$ arcs from the beginning of
+  $r_1$. Moreover, if we factor more than $i$ arcs from the beginning
+  of $r_1$ then the tile corresponding to the last arc in $r_n$ will
+  be visited fewer than $4$ times.
hunk ./TorusAlgebra.tex 484
-\smargin{RL: ``$1\leq i<j\leq n$'' is not consistent with the
-  displayed formula. Later (e.g., statement of
-  Prop~\ref{prop:ATisAinftyPrecise}) seem to repeatedly switch
-  between the two conventions}%
-\smargin{PSO: Fixed the subscripts on the $\mu$. Where are we switching between two conventions?}
hunk ./TorusAlgebra.tex 505
-    a(r_{k+1})\neq 0$. Then $\mu^2_{k,2}\neq 0$ and the only other
+    a(r_{k+1})\neq 0$. Then $\mu^2_{k,k+1}\neq 0$ and the only other
hunk ./TorusAlgebra.tex 509
-    $\mu^2_{1,j}=\mu^2_{i,n}\neq 0$
+    $\mu^2_{1,j}=\mu^2_{i,n}\neq 0$.
hunk ./TorusAlgebra.tex 563
-  where the tile $T$ associated to the first arc in $r_1''$ is %
-  \smargin{RL: does ``agrees with the tile'' mean ``is the same as''?}%
-  \smargin{PSO: I agree with you, but I guess I'm not the same as you. Changed.} %
+  where the tile $T$ associated to the first arc in $r_1''$ is 
hunk ./TorusAlgebra.tex 576
-  Let $r_1,\dots,r_n$ be arcs in a weakly  %
-  \smargin{RL: is the ``strongly admissible'' case also allowed here?}%
-  \smargin{PSO: Yes. Changed statement to use less clumsy  ``weakly admissible''}
-  admissible sequence. Suppose that $a_i$ and $a_j$ with $i<j$ are
+  Let $r_1,\dots,r_n$ be a weakly admissible sequence of Reeb
+  chords. Suppose that $a_i$ and $a_j$ with $i<j$ are 
hunk ./TorusAlgebra.tex 585
-    \smargin{RL: verb in this bullet item?}
-    \smargin{PSO: added a few words}
-  \item If $b$ is some arc which is joined to the immediate left of %
-    \smargin{RL: what does ``joined'' mean?}%
-    \smargin{PSO: hmmm... Let's talk about this}
-    $a_j$ in our sequence, then it is the fourth arc associated to
-    some tile whose first arc $c$ is joined to the immediate right of $a_i$.
+  \item If there is an arc $b$ which is juxtaposed to the immediate left
+    of $a_j$ in (one of the Reeb chords in) our sequence, then $b$ is
+    the fourth arc associated to some tile whose first arc $c$ is
+    juxtaposed to the immediate right of $a_i$.
hunk ./TorusAlgebra.tex 733
-      \smargin{RL: what does ``indeed'' mean here? ``further''? Or is
-        the second clause supposed to imply the first?}%
-      \smargin{PSO: Replaced ``indeed'' by ``moreover''}
hunk ./TorusAlgebra.tex 749
-  \begin{enumerate}[label=(j-\arabic*),ref=j-arabic*]
+  \begin{enumerate}[label=(j-\arabic*),ref=j-\arabic*]