Update to v0.1:v0.1

Changed: Slight change in the statement of the main theorem: no long-range
         positional order between any dimers, regardless of orientation.

Changed: Section 4 was completely reworked. It now consists of a
         discussion, in which definitions and figures appear as needed.
         The main lemma of the section was moved to the end.

Fixed: The definition of external contours had a serious typo in it.

Fixed: The definition of non-trivial polymers was backwards. What were
       called non-trivial polymers are actually trivial, and vice versa.

Changed: The contour and polymer figures were changed to include an
         example of a polymer with a non-trivial recursive structure.

Fixed: The clusters appearing in cluster expansions must be defined using
       multisets instead of sets, as they were before. The notation was
       adjusted accordingly.

Fixed: The expression of the Ursell function was missing a combinatorial
       factor.

Fixed: When splitting a sum over clusters into a sum over a polymer and
       the rest of the cluster, the sum over the multiplictity of the
       polymer must be separated as well, in order to avoid overcounting.

Fixed: In the proof of lemma 5.2, the bound on \mathfrak x required a
       better estimate on b_+ and nu_+.

Fixed: In the proof of lemma 5.2, the argument for why the number of bad
       edges on the boundary of \iota is bounded by \ell_0(l-m) was wrong.

Added: In the proof of lemma 5.2, a figure was added to illustrate the
       issue with bad edges.

Added: Some more minor points have been expanded to include more details.

Fixed: Miscellaneous typos and reformating.

10 files changed, 164 insertions, 29 deletions

diff --git a/figs/dimer_example.fig/Makefile b/figs/dimer_example.fig/Makefile
index 8ed4891..b5e148e 100644
--- a/figs/dimer_example.fig/Makefile
+++ b/figs/dimer_example.fig/Makefile
@@ -1,4 +1,4 @@
-PROJECTNAME=interaction boundary bounding_loop dimer_contour
+PROJECTNAME=interaction boundary bounding_loop dimer_loops dimer_contour
 LIBS=$(notdir $(wildcard libs/*))
 
 PDFS=$(addsuffix .pdf, $(PROJECTNAME))
diff --git a/figs/dimer_example.fig/dimer_conf.py b/figs/dimer_example.fig/dimer_conf.py
index 7cff9f1..a570b91 100644
--- a/figs/dimer_example.fig/dimer_conf.py
+++ b/figs/dimer_example.fig/dimer_conf.py
@@ -274,6 +274,14 @@ print("\\grid{"+str(4*l0+3)+"}{"+str(13)+"}{("+str(mu-2*l0-2)+","+str(mu-7)+")}\
 draw_loops([loop1],bounding_loop,"black")
 close_tikz(bounding_loop)
 
+loops=init_tikz("dimer_loops.tikz.tex")
+print("\\grid{"+str(L-1)+"}{"+str(L-1)+"}{(0,0)}\n", file=loops)
+draw_loops([loop1,loop2,loop3],loops,"blue")
+draw_interactions(v_interactions,h_interactions,loops)
+draw_dimers(v_dimers,h_dimers,loops,"black")
+draw_dimers(v_mantle,h_mantle,loops,"gray")
+close_tikz(loops)
+
 contour=init_tikz("dimer_contour.tikz.tex")
 print("\\draw[color=green, line width=15pt]("+str(mu-2-0.5)+","+str(mu+1)+")--++(1,0);", file=contour)
 print("\\draw[color=green, line width=15pt]("+str(mu-2-0.5)+","+str(mu+2)+")--++(1,0);", file=contour)
diff --git a/figs/polymer_example.fig/contours.tikz.tex b/figs/polymer_example.fig/contours.tikz.tex
index c49c0cc..995bda2 100644
--- a/figs/polymer_example.fig/contours.tikz.tex
+++ b/figs/polymer_example.fig/contours.tikz.tex
@@ -20,9 +20,6 @@
 % magenta
 \fill[color=blue](13,21.5)--++(18,0)--++(0.5,0.5)--++(0,7)--++(-0.5,0.5)--++(-18,0)--++(-0.5,-0.5)--++(0,-7)--cycle;
 \fill[color=white](14,23.5)--++(16,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-16,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
-% cyan
-\fill[color=blue](23,11.5)--++(7,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-7,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
-\fill[color=white](24,13.5)--++(5,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
 % orange
 \fill[color=blue](15,23.5)--++(5,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
 \fill[color=white](17,24.5)--++(1,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-1,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
@@ -37,10 +34,6 @@
   \draw[color=red, line width=15pt](10.5,\i)--++(3,0);
 }
 
-\foreach \i in {12,...,17}{
-  \draw[color=blue, line width=15pt](20.5,\i)--++(2,0);
-  \draw[color=blue, line width=15pt](30.5,\i)--++(2,0);
-}
 \foreach \i in {22,...,29}{
   \draw[color=blue, line width=15pt](10.5,\i)--++(2,0);
   \draw[color=blue, line width=15pt](31.5,\i)--++(1,0);
@@ -60,7 +53,6 @@
 \draw[color=black, line width=5pt](6,8.5)--++(4,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-4,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
 \draw[color=black, line width=5pt](9,19.5)--++(9,0)--++(0.5,-0.5)--++(0,-10)--++(0.5,-0.5)--++(15,0)--++(0.5,0.5)--++(0,22)--++(-0.5,0.5)--++(-25,0)--++(-0.5,-0.5)--++(0,-11)--cycle;
 \draw[color=black, line width=5pt](13,21.5)--++(18,0)--++(0.5,0.5)--++(0,7)--++(-0.5,0.5)--++(-18,0)--++(-0.5,-0.5)--++(0,-7)--cycle;
-\draw[color=black, line width=5pt](23,11.5)--++(7,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-7,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
 \draw[color=black, line width=5pt](15,23.5)--++(5,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
 \draw[color=black, line width=5pt](23,24.5)--++(3,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
 
diff --git a/figs/polymer_example.fig/external_polymer.tikz.tex b/figs/polymer_example.fig/external_polymer.tikz.tex
new file mode 100644
index 0000000..48503df
--- /dev/null
+++ b/figs/polymer_example.fig/external_polymer.tikz.tex
@@ -0,0 +1,68 @@
+\documentclass{standalone}
+
+\usepackage{tikz}
+\usepackage{dimer}
+
+\begin{document}
+\begin{tikzpicture}
+
+%% mantles
+
+% blue
+\fill[color=gray](2,6.5)--++(13,0)--++(0.5,0.5)--++(0,8)--++(-0.5,0.5)--++(-13,0)--++(-0.5,-0.5)--++(0,-8)--cycle;
+\fill[color=white](4,7.5)--++(9,0)--++(0.5,0.5)--++(0,6)--++(-0.5,0.5)--++(-9,0)--++(-0.5,-0.5)--++(0,-6)--cycle;
+% red
+\fill[color=gray](6,8.5)--++(4,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-4,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
+\fill[color=white](7,10.5)--++(2,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-2,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
+% teal
+\fill[color=gray](9,19.5)--++(9,0)--++(0.5,-0.5)--++(0,-10)--++(0.5,-0.5)--++(15,0)--++(0.5,0.5)--++(0,22)--++(-0.5,0.5)--++(-25,0)--++(-0.5,-0.5)--++(0,-11)--cycle;
+\fill[color=white](11,20.5)--++(7,0)--++(0.5,-0.5)--++(0.5,-0.5)--++(1,0)--++(0.5,-0.5)--++(0,-9)--++(0.5,-0.5)--++(11,0)--++(0.5,0.5)--++(0,20)--++(-0.5,0.5)--++(-21,0)--++(-0.5,-0.5)--++(0,-9)--cycle;
+% magenta
+\fill[color=gray](13,21.5)--++(18,0)--++(0.5,0.5)--++(0,7)--++(-0.5,0.5)--++(-18,0)--++(-0.5,-0.5)--++(0,-7)--cycle;
+\fill[color=white](14,23.5)--++(16,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-16,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
+% orange
+\fill[color=gray](15,23.5)--++(5,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
+\fill[color=white](17,24.5)--++(1,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-1,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
+% green
+\fill[color=gray](23,24.5)--++(3,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
+
+
+%% segments
+
+\draw[color=gray, line width=15pt](4,-0.5)--++(0,7);
+\draw[color=gray, line width=15pt](10,15.5)--++(0,4);
+\draw[color=gray, line width=15pt](11,15.5)--++(0,4);
+\draw[color=gray, line width=15pt](9,31.5)--++(0,5);
+
+\foreach \i in {9,...,14}{
+  \draw[color=gray, line width=15pt](3.5,\i)--++(2,0);
+  \draw[color=gray, line width=15pt](10.5,\i)--++(3,0);
+}
+
+\foreach \i in {22,...,29}{
+  \draw[color=gray, line width=15pt](10.5,\i)--++(2,0);
+  \draw[color=gray, line width=15pt](31.5,\i)--++(1,0);
+}
+
+\foreach \i in {23,...,26}{
+  \draw[color=gray, line width=15pt](\i,23.5)--++(0,1);
+  \draw[color=gray, line width=15pt](\i,26.5)--++(0,1);
+}
+
+%% grid
+\grid{36}{36}{[color=lightgray](0,0)}
+
+
+%% loops
+
+\draw[color=black, line width=5pt](2,6.5)--++(13,0)--++(0.5,0.5)--++(0,8)--++(-0.5,0.5)--++(-13,0)--++(-0.5,-0.5)--++(0,-8)--cycle;
+\draw[color=black, line width=5pt](6,8.5)--++(4,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-4,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
+
+\draw[color=black, line width=5pt](9,19.5)--++(9,0)--++(0.5,-0.5)--++(0,-10)--++(0.5,-0.5)--++(15,0)--++(0.5,0.5)--++(0,22)--++(-0.5,0.5)--++(-25,0)--++(-0.5,-0.5)--++(0,-11)--cycle;
+\draw[color=black, line width=5pt](13,21.5)--++(18,0)--++(0.5,0.5)--++(0,7)--++(-0.5,0.5)--++(-18,0)--++(-0.5,-0.5)--++(0,-7)--cycle;
+\draw[color=black, line width=5pt](15,23.5)--++(5,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
+\draw[color=black, line width=5pt](23,24.5)--++(3,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/figs/polymer_example.fig/iota_boundary.tikz.tex b/figs/polymer_example.fig/iota_boundary.tikz.tex
new file mode 100644
index 0000000..af2fa37
--- /dev/null
+++ b/figs/polymer_example.fig/iota_boundary.tikz.tex
@@ -0,0 +1,38 @@
+\documentclass{standalone}
+
+\usepackage{tikz}
+\usepackage{dimer}
+
+\begin{document}
+\begin{tikzpicture}
+
+%% mantles
+\fill[color=gray](2,10.5)--++(18,0)--++(0.5,-0.5)--++(0,-4)--++(-0.5,-0.5)--++(-4,0)--++(-0.5,-0.5)--++(0,-3)--++(0.5,-0.5)--++(11,0)--++(0.5,0.5)--++(0,26)--++(-0.5,0.5)--++(-25,0)--++(-0.5,-0.5)--++(0,-17)--cycle;
+\fill[color=white](4,11.5)--++(17,0)--++(1,-1)--++(0.5,-0.5)--++(0,-4)--++(-1.5,-1.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-1)--++(0.5,-0.5)--++(7,0)--++(0.5,0.5)--++(0,24)--++(-0.5,0.5)--++(-21,0)--++(-0.5,-0.5)--++(0,-15)--cycle;
+\fill[color=gray](7,16.5)--++(10,0)--++(0.5,0.5)--++(0,7)--++(-0.5,0.5)--++(-10,0)--++(-0.5,-0.5)--++(0,-7)--cycle;
+\fill[color=white](8,18.5)--++(8,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-8,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
+
+% iota
+\fill[color=cyan](4,11.5)--++(17,0)--++(1.5,-1.5)--++(3,0)--++(0,17)--++(-0.5,0.5)--++(-21,0)--++(-0.5,-0.5)--++(0,-3)--++(3,0)--++(0.5,0.5)--++(10,0)--++(0.5,-0.5)--++(0,-7)--++(-0.5,-0.5)--++(-10,0)--++(-0.5,.5)--++(-3,0)--++(0,-5)--cycle;
+
+%% segments
+\draw[color=red, line width=20pt](3.5,17)--++(3,0);
+\draw[color=red, line width=20pt](3.5,24)--++(3,0);
+\foreach \i in {18,...,23}{
+  \draw[color=gray, line width=15pt](3.5,\i)--++(3,0);
+}
+\draw[color=red, line width=20pt](22.5,10)--++(3,0);
+\foreach \i in {6,...,9}{
+  \draw[color=gray, line width=15pt](22.5,\i)--++(3,0);
+}
+
+%% grid
+\grid{29}{30}{[color=lightgray](0,0)}
+
+%% loops
+\draw[color=black, line width=5pt](2,10.5)--++(18,0)--++(0.5,-0.5)--++(0,-4)--++(-0.5,-0.5)--++(-4,0)--++(-0.5,-0.5)--++(0,-3)--++(0.5,-0.5)--++(11,0)--++(0.5,0.5)--++(0,26)--++(-0.5,0.5)--++(-25,0)--++(-0.5,-0.5)--++(0,-17)--cycle;
+\draw[color=black, line width=5pt](7,16.5)--++(10,0)--++(0.5,0.5)--++(0,7)--++(-0.5,0.5)--++(-10,0)--++(-0.5,-0.5)--++(0,-7)--cycle;
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/figs/polymer_example.fig/lollipops.tikz.tex b/figs/polymer_example.fig/lollipops.tikz.tex
index 6b6a058..e1a9df1 100644
--- a/figs/polymer_example.fig/lollipops.tikz.tex
+++ b/figs/polymer_example.fig/lollipops.tikz.tex
@@ -21,8 +21,7 @@
 \fill[color=teal](13,21.5)--++(18,0)--++(0.5,0.5)--++(0,7)--++(-0.5,0.5)--++(-18,0)--++(-0.5,-0.5)--++(0,-7)--cycle;
 \fill[color=white](14,23.5)--++(16,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-16,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
 % cyan
-\fill[color=cyan](23,11.5)--++(7,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-7,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
-\fill[color=white](24,13.5)--++(5,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
+\fill[color=cyan](25,13.5)--++(3,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
 % orange
 \fill[color=teal](15,23.5)--++(5,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
 \fill[color=white](17,24.5)--++(1,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-1,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
@@ -36,8 +35,6 @@
 
 \draw[color=red, line width=3pt](3.5,12)--++(2,0);
 
-\draw[color=cyan, line width=3pt](20.5,14)--++(2,0);
-
 \foreach \i in {22,...,29}{
   \draw[color=teal, line width=3pt](31.5,\i)--++(1,0);
 }
@@ -47,6 +44,8 @@
   \draw[color=teal, line width=3pt](\i,26.5)--++(0,1);
 }
 
+\draw[color=cyan, line width=3pt](25,13.5)--++(0,-4);
+
 %% grid
 \grid{36}{36}{[color=lightgray](0,0)}
 
@@ -64,8 +63,8 @@
 
 \draw[color=black, line width=1pt](13,21.5)--++(18,0)--++(0.5,0.5)--++(0,7)--++(-0.5,0.5)--++(-18,0)--++(-0.5,-0.5)--++(0,-7)--cycle;
 
-\draw[color=black, line width=1pt](23,11.5)--++(7,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-7,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
-\draw[color=black](23,18.5)node{\it e};
+\draw[color=black, line width=1pt](25,13.5)--++(3,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
+\draw[color=black](23.5,18)node{\it e};
 
 \draw[color=black, line width=1pt](15,23.5)--++(5,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
 
diff --git a/figs/polymer_example.fig/loops.tikz.tex b/figs/polymer_example.fig/loops.tikz.tex
index db06058..a25747f 100644
--- a/figs/polymer_example.fig/loops.tikz.tex
+++ b/figs/polymer_example.fig/loops.tikz.tex
@@ -20,7 +20,7 @@
 \draw[color=magenta, line width=1pt](13,21.5)--++(18,0)--++(0.5,0.5)--++(0,7)--++(-0.5,0.5)--++(-18,0)--++(-0.5,-0.5)--++(0,-7)--cycle;
 \draw[color=black](13,30.5)node{\it d};
 
-\draw[color=cyan, line width=1pt](23,11.5)--++(7,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-7,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
+\draw[color=cyan, line width=1pt](25,11.5)--++(3,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
 \draw[color=black](23,18.5)node{\it e};
 
 \draw[color=orange, line width=1pt](15,23.5)--++(5,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
diff --git a/figs/polymer_example.fig/mbbX.tikz.tex b/figs/polymer_example.fig/mbbX.tikz.tex
new file mode 100644
index 0000000..61db917
--- /dev/null
+++ b/figs/polymer_example.fig/mbbX.tikz.tex
@@ -0,0 +1,31 @@
+\documentclass{standalone}
+
+\usepackage{tikz}
+\usepackage{dimer}
+
+\begin{document}
+\begin{tikzpicture}
+
+%% mantles
+\fill[color=gray](2,1.5)--++(13,0)--++(0.5,0.5)--++(0,8)--++(-0.5,0.5)--++(-13,0)--++(-0.5,-0.5)--++(0,-8)--cycle;
+\fill[color=white](4,2.5)--++(9,0)--++(0.5,0.5)--++(0,6)--++(-0.5,0.5)--++(-9,0)--++(-0.5,-0.5)--++(0,-6)--cycle;
+\fill[color=gray](4,3.5)--++(4,0)--++(0.5,0.5)--++(0,4)--++(-0.5,0.5)--++(-4,0)--++(-0.5,-0.5)--++(0,-4)--cycle;
+\fill[color=white](5,5.5)--++(2,0)--++(0.5,0.5)--++(0,0)--++(-0.5,0.5)--++(-2,0)--++(-0.5,-0.5)--++(0,-0)--cycle;
+
+%% segments
+\foreach \i in {4,...,8}{
+  \draw[color=gray, line width=15pt](8.5,\i)--++(5,0);
+}
+
+%% grid
+\grid{17}{12}{[color=lightgray](0,0)}
+
+
+%% loops
+\draw[color=black, line width=5pt](2,1.5)--++(13,0)--++(0.5,0.5)--++(0,8)--++(-0.5,0.5)--++(-13,0)--++(-0.5,-0.5)--++(0,-8)--cycle;
+\draw[color=black, line width=5pt](3.5,4)--++(0.5,-0.5)--++(4,0)--++(0.5,0.5)--++(0,4)--++(-0.5,0.5)--++(-4,0)--++(-0.5,-0.5);
+\draw[color=red, line width=10pt](3.5,8)--++(0,-4);
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/figs/polymer_example.fig/polymers.tikz.tex b/figs/polymer_example.fig/polymer.tikz.tex
index e2f688a..6c18e7a 100644
--- a/figs/polymer_example.fig/polymers.tikz.tex
+++ b/figs/polymer_example.fig/polymer.tikz.tex
@@ -21,8 +21,7 @@
 \fill[color=gray](13,21.5)--++(18,0)--++(0.5,0.5)--++(0,7)--++(-0.5,0.5)--++(-18,0)--++(-0.5,-0.5)--++(0,-7)--cycle;
 \fill[color=white](14,23.5)--++(16,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-16,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
 % cyan
-\fill[color=gray](23,11.5)--++(7,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-7,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
-\fill[color=white](24,13.5)--++(5,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
+\fill[color=gray](25,13.5)--++(3,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
 % orange
 \fill[color=gray](15,23.5)--++(5,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
 \fill[color=white](17,24.5)--++(1,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-1,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
@@ -42,10 +41,6 @@
   \draw[color=gray, line width=15pt](10.5,\i)--++(3,0);
 }
 
-\foreach \i in {12,...,17}{
-  \draw[color=gray, line width=15pt](20.5,\i)--++(2,0);
-  \draw[color=gray, line width=15pt](30.5,\i)--++(2,0);
-}
 \foreach \i in {22,...,29}{
   \draw[color=gray, line width=15pt](10.5,\i)--++(2,0);
   \draw[color=gray, line width=15pt](31.5,\i)--++(1,0);
@@ -56,6 +51,10 @@
   \draw[color=gray, line width=15pt](\i,26.5)--++(0,1);
 }
 
+\draw[color=gray, line width=15pt](25,13.5)--++(0,-4);
+\draw[color=gray, line width=15pt](27,17.5)--++(0,4);
+
+
 %% grid
 \grid{36}{36}{[color=lightgray](0,0)}
 
@@ -67,7 +66,7 @@
 
 \draw[color=black, line width=5pt](9,19.5)--++(9,0)--++(0.5,-0.5)--++(0,-10)--++(0.5,-0.5)--++(15,0)--++(0.5,0.5)--++(0,22)--++(-0.5,0.5)--++(-25,0)--++(-0.5,-0.5)--++(0,-11)--cycle;
 \draw[color=black, line width=5pt](13,21.5)--++(18,0)--++(0.5,0.5)--++(0,7)--++(-0.5,0.5)--++(-18,0)--++(-0.5,-0.5)--++(0,-7)--cycle;
-\draw[color=black, line width=5pt](23,11.5)--++(7,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-7,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
+\draw[color=black, line width=5pt](25,13.5)--++(3,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
 \draw[color=black, line width=5pt](15,23.5)--++(5,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
 \draw[color=black, line width=5pt](23,24.5)--++(3,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
 
diff --git a/figs/polymer_example.fig/segments.tikz.tex b/figs/polymer_example.fig/segments.tikz.tex
index 9669e3b..b854f54 100644
--- a/figs/polymer_example.fig/segments.tikz.tex
+++ b/figs/polymer_example.fig/segments.tikz.tex
@@ -22,8 +22,8 @@
 \fill[color=magenta](13,21.5)--++(18,0)--++(0.5,0.5)--++(0,7)--++(-0.5,0.5)--++(-18,0)--++(-0.5,-0.5)--++(0,-7)--cycle;
 \fill[color=white](14,23.5)--++(16,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-16,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
 % cyan
-\fill[color=cyan](23,11.5)--++(7,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-7,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
-\fill[color=white](24,13.5)--++(5,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
+\fill[color=cyan](25,11.5)--++(3,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
+\fill[color=white](26,13.5)--++(1,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-1,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
 % orange
 \fill[color=orange](15,23.5)--++(5,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
 \fill[color=white](17,24.5)--++(1,0)--++(0.5,0.5)--++(0,1)--++(-0.5,0.5)--++(-1,0)--++(-0.5,-0.5)--++(0,-1)--cycle;
@@ -71,8 +71,8 @@
   \draw[color=teal, line width=15pt](20.5,\i)--++(12,0);
 }
 \foreach \i in {12,...,17}{
-  \draw[color=teal, line width=15pt](20.5,\i)--++(2,0);
-  \draw[color=teal, line width=15pt](30.5,\i)--++(2,0);
+  \draw[color=teal, line width=15pt](20.5,\i)--++(4,0);
+  \draw[color=teal, line width=15pt](28.5,\i)--++(4,0);
 }
 \foreach \i in {18,...,19}{
   \draw[color=teal, line width=15pt](20.5,\i)--++(12,0);
@@ -85,7 +85,7 @@
 }
 \draw[color=teal, line width=15pt](10.5,30)--++(22,0);
 
-\foreach \i in {24,...,29}{
+\foreach \i in {26,...,27}{
   \draw[color=cyan, line width=15pt](\i,13.5)--++(0,2);
 }
 
@@ -120,7 +120,7 @@
 % magenta
 \draw[color=black, line width=5pt](13,21.5)--++(18,0)--++(0.5,0.5)--++(0,7)--++(-0.5,0.5)--++(-18,0)--++(-0.5,-0.5)--++(0,-7)--cycle;
 % cyan
-\draw[color=black, line width=5pt](23,11.5)--++(7,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-7,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
+\draw[color=black, line width=5pt](25,11.5)--++(3,0)--++(0.5,0.5)--++(0,5)--++(-0.5,0.5)--++(-3,0)--++(-0.5,-0.5)--++(0,-5)--cycle;
 % orange
 \draw[color=black, line width=5pt](15,23.5)--++(5,0)--++(0.5,0.5)--++(0,3)--++(-0.5,0.5)--++(-5,0)--++(-0.5,-0.5)--++(0,-3)--cycle;
 %green