diff --git a/papers/plane_depth/paper.aux b/papers/plane_depth/paper.aux
index 6c7a65e..b0d5fa7 100644
--- a/papers/plane_depth/paper.aux
+++ b/papers/plane_depth/paper.aux
@@ -5,18 +5,18 @@
 \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Definitions}}{1}{}\protected@file@percent }
 \newlabel{def:plane-depth}{{1.1}{1}}
 \newlabel{def:level-edge}{{1.2}{1}}
-\newlabel{def:triangle-types}{{1.3}{1}}
+\newlabel{def:level-dual-edge}{{1.3}{1}}
+\newlabel{def:triangle-types}{{1.4}{1}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Outerplanarity of depth levels}}{2}{}\protected@file@percent }
 \newlabel{lem:outerplanarity}{{2.1}{2}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Deep embedding}}{2}{}\protected@file@percent }
 \newlabel{def:deep-embedding}{{3.1}{2}}
-\newlabel{lem:up-down-faces}{{3.2}{2}}
-\bibcite{baker1994}{1}
+\newlabel{lem:up-down-faces}{{3.2}{3}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Quadrilateral decomposition}}{3}{}\protected@file@percent }
 \newlabel{lem:unique-level-edge}{{4.1}{3}}
 \newlabel{lem:shared-level-edge}{{4.2}{3}}
 \newlabel{def:quad-decomposition}{{4.3}{3}}
-\newlabel{def:quad-types}{{4.5}{3}}
+\bibcite{baker1994}{1}
 \bibcite{bauerfeld-pds-seq}{2}
 \bibcite{bauerfeld-nested-tires}{3}
 \newlabel{tocindent-1}{0pt}
@@ -24,5 +24,6 @@
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
+\newlabel{def:quad-types}{{4.5}{4}}
 \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent }
 \gdef \@abspage@last{4}
diff --git a/papers/plane_depth/paper.log b/papers/plane_depth/paper.log
index 0526a94..1f066f0 100644
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diff --git a/papers/plane_depth/paper.pdf b/papers/plane_depth/paper.pdf
index 6bb91c5..8cb4405 100644
Binary files a/papers/plane_depth/paper.pdf and b/papers/plane_depth/paper.pdf differ
diff --git a/papers/plane_depth/paper.tex b/papers/plane_depth/paper.tex
index 928e065..5fed3c5 100644
--- a/papers/plane_depth/paper.tex
+++ b/papers/plane_depth/paper.tex
@@ -76,7 +76,23 @@ $G$.
 \begin{definition}
 \label{def:level-edge}
 An edge $\{u, v\} \in E(G)$ is a \emph{level edge} if
-$\mathrm{depth}(u) = \mathrm{depth}(v)$.
+$\mathrm{depth}(u) = \mathrm{depth}(v)$, and an \emph{interlevel
+edge} if $\mathrm{depth}(u) \neq \mathrm{depth}(v)$.
+\end{definition}
+
+Under planar duality, each primal edge of $G$ corresponds to a
+unique edge of the planar dual --- the edge between the two face
+vertices on either side of the primal edge.  Levels and interlevels
+lift to the dual:
+
+\begin{definition}
+\label{def:level-dual-edge}
+A dual edge of $G$ is a \emph{level dual edge} if it crosses a level
+edge of $G$ in the planar embedding (equivalently, if its
+corresponding primal edge has equal-depth endpoints), and an
+\emph{interlevel dual edge} if it crosses an interlevel edge
+(equivalently, if its corresponding primal edge has
+different-depth endpoints).
 \end{definition}
 
 \begin{definition}