crossing overs in fig. 4.3 and 4.4

group.tex

@@ -1896,10 +1896,10 @@ \section{The sign homomorphism}
       \node[dot,label=below:{$\n$}] (y\n) at (\n,0) {};
     }
     \draw[->] (x1) -- (y2);
-    \draw[->] (x2) -- (y3);
-    \draw[->] (x3) -- (y1);
-    \draw[->] (x4) -- (y5);
-    \draw[->] (x5) -- (y4);
+    \draw[->,commutative diagrams/crossing over] (x2) -- (y3);
+    \draw[->,commutative diagrams/crossing over] (x3) -- (y1);
+    \draw[->,commutative diagrams/crossing over] (x4) -- (y5);
+    \draw[->,commutative diagrams/crossing over] (x5) -- (y4);
   \end{tikzpicture}
   \caption{A different representation of the permutation $\sigma$
     from~\cref{fig:cycle-decomposition}.}
@@ -1937,10 +1937,10 @@ \section{The sign homomorphism}
       \node[dot,gray!25] (z3) at (3,1) {};
       \draw[->] (x0) -- (z1);
       \draw[->] (z1) -- (y0);
-      \draw[->] (x1) -- (z0);
-      \draw[->] (z0) -- (y1);
+      \draw[->,commutative diagrams/crossing over] (x1) -- (z0);
+      \draw[->,commutative diagrams/crossing over] (z0) -- (y1);
       \draw[->] (x2) .. controls (3,1) and (3,1) .. (y2);
-      \draw[->] (x3) .. controls (2,1) and (2,1) .. (y3);
+      \draw[->,commutative diagrams/crossing over] (x3) .. controls (2,1) and (2,1) .. (y3);
       \draw[->] (x4) -- (y4);
       \draw[->] (x5) -- (y5);
     \end{tikzpicture}