Cross over of arrows in a complex diagram

With crossing over, arrows that are drawn later, erase part of the arrows they cross.

\documentclass{article}
\usepackage{geometry}
\usepackage{amsmath}

\usepackage{tikz-cd}

\newcommand{\level}{\mathrm{Level}}

\begin{document}

\begin{tikzcd}[column sep=1.5em,row sep=large,arrows={crossing over}]
&&
\level(1,3)
\\ \\
&
\level(2,2) \arrow[uur] &
\level(2,3) \arrow[uu] \arrow[rr,thick,equals] &&
\level(2,5) \arrow[rr] \arrow[ddl,leftarrow] &&
\level(2,7) \arrow[ddl,thick,equals]
\\
&&
\level(3,3) \arrow[u] \arrow[rr] \arrow[ddl,leftarrow] &&
\level(3,5) \arrow[u,thick,equals] \arrow[rr] \arrow[ddl,leftarrow] &&
\level(3,7) \arrow[u,thick,equals]
\\
\level(4,1) \arrow[uur,thick,equals] &
\level(4,2) \arrow[uu] \arrow[rr] \arrow[uur] &&
\level(4,4) \arrow[rr] \arrow[ddl,leftarrow] &&
\level(4,6) \arrow[ddl,leftarrow]
\\
&
\level(5,2) \arrow[u] \arrow[rr] \arrow[ddl,leftarrow] &&
\level(5,4) \arrow[u] \arrow[rr] &&
\level(5,6) \arrow[u,thick,equals] \arrow[uur]
\\
\level(6,1) \arrow[uu] \arrow[uur,thick,equals] \arrow[rr] &&
\level(6,3) \arrow[rr] &&
\level(6,5)
\\
\level(7,1) \arrow[u] \arrow[rr] &&
\level(7,3) \arrow[u] \arrow[rr] &&
\level(7,5) \arrow[u] \arrow[uur]
\end{tikzcd}

\end{document} 

The standard trick for this is to draw a thick white line underneath, which can be achieved with preaction. This answer comes with a style on top, which can be used like (m-6-5) edge[on top] (m-4-6). You need to draw the topmost connections last. At the moment, the default line widths of the white lines is 4pt but you can adjust them to your needs.

\documentclass[border=5mm]{standalone}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{matrix}
\begin{document}
\begin{tikzpicture}[on top/.style={preaction={draw=white,-,line width=#1}},
on top/.default=4pt]
  \matrix (m) [matrix of math nodes, row sep=1.6cm,
    column sep=0.4cm]{
    &&Level(1,3)&&&&\\
     &Level(2,2)&Level(2,3)&& Level(2,5)&&Level(2,7)\\
     &&Level(3,3)&& Level(3,5)&&Level(3,7)\\
      Level(4,1)&Level(4,2)&& Level(4,4)&&Level(4,6)&  \\
      &Level(5,2)&& Level(5,4)&&Level(5,6)& \\
       Level(6,1)&& Level(6,3)&& Level(6,5)&&\\
        Level(7,1) && Level(7,3)&& Level(7,5)&&\\};
  \path[-stealth]
(m-7-1) edge (m-7-3) edge (m-6-1)

(m-7-3) edge (m-7-5) 
        edge (m-6-3)

(m-7-5) edge (m-6-5) 
        edge (m-5-6)

(m-6-1) edge (m-4-1)

(m-6-3) edge (m-6-5) 

(m-5-4) edge (m-4-4) 

(m-5-2) edge (m-5-4) 
        edge (m-4-2)
        edge (m-3-3)

(m-5-4)   edge (m-5-6)
        edge (m-3-5)
(m-5-6) edge (m-3-7)
(m-4-2)  edge (m-2-2) 
        edge (m-2-3)



(m-3-3) edge (m-2-3) 
        edge (m-3-5)

 (m-3-5) edge (m-3-7)
 (m-2-2) edge (m-1-3)
 (m-2-3) edge (m-1-3)
 (m-2-5) edge (m-2-7)
(m-4-2) edge[on top] (m-4-4) 
(m-4-4) edge[on top] (m-4-6) 
(m-6-5) edge[on top] (m-4-6)
(m-4-4) edge[on top] (m-2-5)
(m-6-3) edge[on top] (m-4-4)
 ;

\path[>=stealth,-,thick]
(m-7-1) edge [double] (m-5-2)
(m-6-1) edge [double] (m-4-2)
(m-5-6) edge [double] (m-4-6)
(m-4-1) edge [double] (m-2-2)
(m-3-5) edge [double] (m-2-5)
(m-3-7) edge [double] (m-2-7)
(m-2-3) edge [double] (m-2-5)
(m-4-6) edge [double,on top=5pt] (m-2-7)
;
\path[-stealth] (m-6-1) edge[on top] (m-6-3) ;
\end{tikzpicture}
\end{document} 

@egreg called this code "awkward". The IMHO most redundant thing, however, is IMHO that one has to add the position of the entry by hand (but I do not like to use words like "awkward"). In any case, this can be made as easily as defining a style

level/.style={
execute at begin node={\text{Level}(\the\pgfmatrixcurrentrow,\the\pgfmatrixcurrentcolumn)}
}

which gets used in the following code, which also has some loops to draw the arrows.

\documentclass[border=5mm]{standalone}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{matrix}
\begin{document}
\begin{tikzpicture}[on top/.style={preaction={draw=white,-,line width=#1}},
on top/.default=4pt,level/.style={
execute at begin node={\text{Level}(\the\pgfmatrixcurrentrow,\the\pgfmatrixcurrentcolumn)}
}]
  \matrix (m) [matrix of math nodes,nodes=level,row sep=1.6cm,
    column sep=0.4cm]{
    &&{}&&&&\\
     &{}&{}&& {}&&{}\\
     &&{}&& {}&&{}\\
      {}&{}&& {}&&{}&  \\
      &{}&& {}&&{}& \\
       {}&& {}&& {}&&\\
        {} && {}&& {}&&\\};
\path[-stealth] 
 (m-6-1) edge (m-4-1) (m-4-2) edge (m-2-2) (m-2-3) edge (m-1-3)
 (m-4-2) edge (m-2-3) (m-5-2) edge (m-3-3) 
 (m-5-4) edge (m-3-5) (m-7-5) edge (m-5-6) (m-5-6) edge (m-3-7)
 (m-2-2) edge (m-1-3);
\path[-stealth,every edge/.append style={on top}]  
foreach \Y [count=\Z] in {3,5,7}{ 
  foreach \X in {1,3,5}
 {(m-\Y-\the\numexpr\X+3-\Z\relax) edge (m-\the\numexpr\Y-1\relax-\the\numexpr\X+3-\Z\relax)
 \ifnum\X<5
 (m-\Y-\the\numexpr\X+3-\Z\relax) edge (m-\Y-\the\numexpr\X+5-\Z\relax)
 \ifnum\the\numexpr\Y*10+\X=31
 \else
 (m-\the\numexpr\Y-1\relax-\the\numexpr\X+3-\Z\relax) edge (m-\the\numexpr\Y-1\relax-\the\numexpr\X+5-\Z\relax)
 \fi
 \fi
 }} (m-4-4) edge (m-2-5) (m-6-3) edge (m-4-4);
 \path[thick,every edge/.append style={double,on top=5pt}] 
  (m-4-1) edge (m-2-2) (m-6-1) edge (m-4-2) (m-7-1) edge (m-5-2)
  (m-2-3) edge (m-2-5) (m-2-7) edge (m-3-7)
  (m-2-7) edge (m-4-6) (m-4-6) edge (m-6-5);
\path[-stealth,every edge/.append style={on top}]  (m-6-1) edge (m-6-3); 
\end{tikzpicture}
\end{document} 

This information, together with this answer, allows one to simplify egreg's answer and to correct the arrows there (such that there is a 3D impression).

\documentclass{article}
\usepackage{geometry}
\usepackage{amsmath}
\usepackage{tikz-cd}
\usetikzlibrary{backgrounds}

\newcommand{\level}{\text{level}(\the\pgfmatrixcurrentrow,\the\pgfmatrixcurrentcolumn)}

\begin{document}

\begin{tikzcd}[column sep=1.5em,row sep=large,arrows={crossing over},
execute at end picture={
\begin{scope}[on background layer]
 \path[->] (\tikzcdmatrixname-6-2) edge (\tikzcdmatrixname-6-4)
 (\tikzcdmatrixname-6-4) edge (\tikzcdmatrixname-6-6)
 (\tikzcdmatrixname-4-5) edge (\tikzcdmatrixname-4-7);
\end{scope}}]
&&
\level
\\ \\
&
\level \arrow[uur] &
\level \arrow[uu] \arrow[rr,thick,equals] &&
\level \arrow[rr] \arrow[ddl,leftarrow] &&
\level \arrow[ddl,thick,equals]
\\
&&
\level \arrow[u] \arrow[ddl,leftarrow] &&
\level \arrow[u,thick,equals]  \arrow[ddl,leftarrow] &&
\level \arrow[u,thick,equals]
\\
\level \arrow[uur,thick,equals] &
\level \arrow[uu] \arrow[rr] \arrow[uur] &&
\level \arrow[rr] \arrow[ddl,leftarrow] &&
\level \arrow[ddl,leftarrow]
\\
&
\level \arrow[u] \arrow[ddl,leftarrow] &&
\level \arrow[u]  &&
\level \arrow[u,thick,equals] \arrow[uur]
\\
\level \arrow[uu] \arrow[uur,thick,equals] \arrow[rr] &&
\level \arrow[rr] &&
\level
\\
\level \arrow[u] \arrow[rr] &&
\level \arrow[u] \arrow[rr] &&
\level \arrow[u] \arrow[uur]
\end{tikzcd}
\end{document}