# Bending snakes with xy-pic

This is a follow-up to this question: How do you draw the "snake" arrow for the connecting homomorphism in the snake lemma?

I'm trying to emulate this snake diagram Aluffi does in his book "Algebra, Chapter 0" with xy-pic:

but I can't modify correctly Matsaya's code to this avail. What I especially like about this diagram is the snake, crossing the diagram that way.

-
Could you post what you have so far? Would you be interested in a PSTricks or a TikZ answer? – cmhughes Oct 15 '11 at 21:22
@cmhughes: I never used it, but I'm of course willing to try it, and it's interesting for everybody. – lentic catachresis Oct 15 '11 at 21:43
It's normal you can't modify my code to do your figure, because the xy-syntaxe '^dl[dl] only permits you to draw 1/8 of circle, but you want 1/16 of circle (^dll). I will think about another way to achieve your desires. – Matsaya Oct 17 '11 at 8:20

Here is a try.

First, the code :

\documentclass{minimal}
\usepackage[all,cmtip]{xy}
\usepackage{amsmath}
\DeclareMathOperator{\coker}{coker}
\newcommand*\pp{{\rlap{$$'$$}}}
\begin{document}
%
$\xymatrix@!R{ & 0 & 0 & 0 \\ 0 \ar@{-->}[r] & {\ker(a)} \ar[r] & {\ker(b)} \ar[r] & {\ker(c)} \ar@{-}r[d][d]^\delta[d] % curved arrow 1 & \\ 0 \ar@{-->}[r] & A \ar[r]^{f} & B \ar[r]^{g} & C \ar[r] \ar@{}+<0.6cm,0cm>="p1" %intersection point 1 & 0 \\ 0 \ar[r] & A\pp \ar[r]^{f'} \ar@{}[l]+<1cm,0cm>="p2" %intersection point 2 \ar_l[l]+<1.03cm,0cm>[d][d][d] % curved arrow 2 & B\pp \ar[r]^{g'} & C\pp \ar@{-->}[r] & 0 \\ & {\coker(a)} \ar[r] & {\coker(b)} \ar[r] & {\coker(c)} \ar@{-->}[r] & 0 \\ & 0 & 0 & 0 \\ % vertical arrows \ar"1,2";"2,2" \ar"1,3";"2,3" \ar"1,4";"2,4" \ar"2,2";"3,2" \ar"2,3";"3,3" \ar"2,4";"3,4" \ar"3,2";"4,2"^a \ar"3,3";"4,3"^<<b \ar"3,4";"4,4"^c \ar"4,2";"5,2" \ar"4,3";"5,3" \ar"4,4";"5,4" \ar"5,2";"6,2" \ar"5,3";"6,3" \ar"5,4";"6,4" % diagonal arrow, with 1 hole \ar@{-}"p1";"p2"|!{"2,3";"3,3"}\hole }$
\end{document}


Then some explanations.

• \xymatrix@!R forces all colon spaces equal

• \newcommand*\pp{{\rlap{$$'$$}}} is to hide the prime's width (not really necessary)

• \ar"1,3";"2,3" is an explicit position of arrow

• |!{[];[d]}\hole makes a hole in the present arrow at the position where it intersects the (straight) line from [] to [d]

• \ard[t1][t2][t3] begins in the d direction, go to the t1 entry, makes a quarter of turn, go to the t2 entry, makes a quarter of turn and ends in t3. The /4pt changes the radius. It's even possible to decide the direction of the turn with ^d[t] or _ul[t] syntax. But you are only able to do 1/8 of turn, not 1/16 as you want.

• Finally, we name the two intersection position p1 and p2 with \ar@{}+<0.6cm,0cm>="p1"(the finding of intersection points is done manually, but it's maybe improvable) and draw an arrow between the two points.

This version is "improved". I.e. the spacing and positioning is more consistent with your example (but the first code could be changed quite easily). I also added dashed arrow @{-->}(you could have dotted with @{..>}) and put the ba little upper (with ^<<b).

You could also do it in a slightly different way and without naming points. To have the slanted arrow, simply go to the left, and then modify the target by an appropriate value (here find by hand too). N.B. I think you only can modify the last entry of a path with the syntax +vector. So you have to draw you arrow in two times.

$\xymatrix@!{ & {\ker(a)} \ar[r] & {\ker(b)} \ar[r] & {\ker(c)} \ar@{-}r[d][d]^\delta[dll]-<1.35cm,2.315cm>|!{"2,3";"3,3"}\hole % curved arrow 1 & \\ & A \ar[r]^{f} & B \ar[r]^{g} & C \ar[r] & 0 \\ 0 \ar[r] & A\pp \ar[r]^{f'} \ar_l[l]+<1cm,0cm>[d][d][d]% curved arrow 2 & B\pp \ar[r]^{g'} & C\pp & \\ & {\coker(a)} \ar[r] & {\coker(b)} \ar[r] & {\coker(c)} & \\ % vertical arrows \ar"1,2";"2,2" \ar"1,3";"2,3" \ar"1,4";"2,4" \ar"2,2";"3,2"^a \ar"2,3";"3,3"^b \ar"2,4";"3,4"^c \ar"3,2";"4,2" \ar"3,3";"4,3" \ar"3,4";"4,4" }$


For more details, see the xy-pic user's guide.

The first example :

The second one :

-
This is great too, and with xypic! The only detail is that I was ambiguous in my comment to Andrew Stacey's answer, in that the $0\to ker(a)$ and $coker(c)\to 0$ arrows that should be dotted are the horizontal ones ;) – lentic catachresis Oct 17 '11 at 17:01
@BrunoStonek it's corrected, but I'm ashamed of my fault ! – Matsaya Oct 17 '11 at 17:32
Very nice! (I fixed a couple of indentation errors in the code blocks.) – Loop Space Oct 17 '11 at 18:08

Following the comments, the OP said he might be interested in a PSTricks solution. Below is a solution using the pst-node package. The 'snake' curve isn't perfect- any ideas for improvement are welcome.

\documentclass{article}

\usepackage{pst-node}

\begin{document}

\psset{xunit=1.75cm,yunit=1cm,nodesep=3pt}
\begin{pspicture}(-1,-1)(5,6)
%\psgrid    % useful during construction
% put the nodes in- working from the bottom upwards
% bottom row
\psnode(1,0){10}{$0$}
\psnode(2,0){20}{$0$}
\psnode(3,0){30}{$0$}
% second row
\psnode(1,1){cokerl}{coker $\lambda$}
\psnode(2,1){cokerm}{coker $\mu$}
\psnode(3,1){cokern}{coker $\nu$}
% third row
\psnode(0,2){02}{$0$}
\psnode(1,2){L0}{$L_0$}
\psnode(2,2){M0}{$M_0$}
\psnode(3,2){N0}{$N_0$}
\psnode(4,2){42}{$0$}
% fourth row
\psnode(0,3){03}{$0$}
\psnode(1,3){L1}{$L_1$}
\psnode(2,3){M1}{$M_1$}
\psnode(3,3){N1}{$N_1$}
\psnode(4,3){43}{$0$}
% fifth row
\psnode(0,4){04}{$0$}
\psnode(1,4){kerl}{ker $\lambda$}
\psnode(2,4){kerm}{ker $\mu$}
\psnode(3,4){kern}{ker $\nu$}
% sixth row
\psnode(1,5){15}{$0$}
\psnode(2,5){25}{$0$}
\psnode(3,5){35}{$0$}
% horizontal arrows
% 2nd row
\ncline{->}{cokerl}{cokerm}
\ncline{->}{cokerm}{cokern}
% 3rd row
\ncline{->}{02}{L0}
\ncline{->}{L0}{M0}
\nbput{$\alpha_0$}
\ncline{->}{M0}{N0}
\nbput{$\beta_0$}
\ncline{->}{N0}{42}
% 4th row
\ncline{->}{03}{L1}
\ncline{->}{L1}{M1}
\naput{$\alpha_1$}
\ncline{->}{M1}{N1}
\naput{$\beta_1$}
\ncline{->}{N1}{43}
% 5th row
\ncline{->}{04}{kerl}
\ncline{->}{kerl}{kerm}
\ncline{->}{kerm}{kern}
% vertical arrows
\ncline{->}{cokerl}{10}
\ncline{->}{cokerm}{20}
\ncline{->}{cokern}{30}
\ncline{->}{L0}{cokerl}
\ncline{->}{M0}{cokerm}
\ncline{->}{N0}{cokern}
\ncline{->}{L1}{L0}
\nbput{$\lambda$}
\ncline{->}{M1}{M0}
\nbput{$\mu$}
\ncline{->}{N1}{N0}
\nbput{$\nu$}
\ncline{->}{kerl}{L1}
\ncline{->}{kerm}{M1}
\ncline{->}{kern}{N1}
\ncline{->}{15}{kerl}
\ncline{->}{25}{kerm}
\ncline{->}{35}{kern}
\nccurve[angleA=0,angleB=180]{->}{kern}{cokerl}
% npos takes a value between 0 and 1 for \nccurve
\naput[npos=0.1]{$\delta$}
\end{pspicture}

\end{document}


And if you change the line \nccurve... and \naput... to

\ncloop[angleA=0,armA=1cm,armB=1.5cm,angleB=180,loopsize=1.5,linearc=0.25]{->}{kern}{cokerl}
\naput[npos=0.4]{$\delta$}


then you get

EDIT

If you want to run the code with pdflatex then your preamble should look like this:

\documentclass{article}

\usepackage[pdf]{pstricks}
\usepackage{pst-node}

\begin{document}
....


You should then run

 pdflatex -shell-escape myfile.tex


If you want to use the code 'as is', then you should either use xelatex myfile.tex or else

 latex myfile.tex
dvips myfile.dvi
ps2pdf myfile.ps

-
I'm sorry, but I'm new to this package, and with your minimal example I'm getting "undefined control sequence" on \psnode. What's going on? – lentic catachresis Oct 16 '11 at 2:03
@BrunoStonek No problem. You can not compile it with pdflatex. You have a few choices. Either you need to use XeLaTeX or else latex=>dvi=>ps=>pdf or else see this post: tex.stackexchange.com/questions/8413/… – cmhughes Oct 16 '11 at 2:20
Thank you for your assistance. After adding \usepackage{auto-pst-pdf} and running pdflatex with -shell-escape I get "Cannot determine size of graphic in a.pdf (no BoundingBox)." – lentic catachresis Oct 16 '11 at 2:44
@BrunoStonek I've added details to the end of my example, hope it helps. – cmhughes Oct 16 '11 at 2:53
I've closely followed your instructions for use with pdflatex but I still get the aformentioned error. – lentic catachresis Oct 16 '11 at 3:03

Here's a modification of the TikZ version from How do you draw the "snake" arrow for the connecting homomorphism in the snake lemma?. One could fine tune the spacing and curving a bit better, I guess. Also, if I were writing it from fresh I'd label things more appropriately for this incantation of the snake lemma.

Code:

\documentclass{article}
%\url{http://tex.stackexchange.com/q/31687/86}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{%
matrix,%
calc,%
arrows%
}

\DeclareMathOperator{\coker}{coker}

\begin{document}
\begin{tikzpicture}[>=angle 60]
\matrix[matrix of math nodes,column sep={30pt},row sep={40pt,between origins}] (s)
{
&|[name=0a]| 0 &|[name=0b]| 0 &|[name=0c]| 0 \\
%
|[name=0k]| 0 &|[name=ka]| \ker \lambda &|[name=kb]| \ker \mu &|[name=kc]| \ker \nu \\
%
|[name=0A]| 0 &|[name=A]| L_1 &|[name=B]| M_1 &|[name=C]| N_1 &|[name=01]| 0 \\
%
|[name=02]| 0 &|[name=A']| L_0 &|[name=B']| M_0 &|[name=C']| N_0 &|[name=0C']| 0 \\
%
&|[name=ca]| \coker \lambda &|[name=cb]| \coker \mu &|[name=cc]| \coker \nu &|[name=0ck]| 0 \\
%
&|[name=0ca]| 0 &|[name=0cb]| 0 &|[name=0cc]| 0 \\
%
};

% Horizontal arrows
\foreach \start/\end in {%
0A/A,
A/B,
B/C,
C/01,
02/A',
A'/B',
B'/C',
C'/0C',
0k/ka,
ka/kb,
kb/kc,
ca/cb,
cb/cc,
cc/0ck%
} {
\draw[->] (\start.mid east) -- (\end.mid west);
}

% Vertical arrows
\foreach \start/\end in {%
0a/ka,
0b/kb,
0c/kc,
ka/A,
kb/B,
kc/C,
A/A',
B/B',
C/C',
A'/ca,
B'/cb,
C'/cc,
ca/0ca,
cb/0cb,
cc/0cc%
} {
\draw[->] (\start) -- (\end);
}

\path (A.mid east) -- node[auto] {$$\alpha_1$$} (B.mid west);
\path (B.mid east) -- node[auto] {$$\beta_1$$} (C.mid west);
\path (A'.mid east) -- node[auto,swap] {$$\alpha_0$$} (B'.mid west);
\path (B'.mid east) -- node[auto,swap] {$$\beta_0$$} (C'.mid west);
\path (A) -- node[auto,swap] {$$\lambda$$} (A');
\path (B) -- node[auto,swap,pos=.3] {$$\mu$$} (B');
\path (C) -- node[auto,swap] {$$\nu$$} (C');

\path (C.mid east) -- node (gap) {} (A'.mid west);

\draw[->] (kc.mid east) to[out=0,in=90] ($(kc)+(1,-.5)$) -- node[auto] {$$\delta$$} ($(C)+(1,.5)$) to[out=-90,in=0] (C.mid east) -- (gap) -- (A'.mid west) to[out=180,in=90] ($(A')+(-1.3,-.5)$) -- ($(ca)+(-1.3,.5)$) to[out=-90,in=180] (ca.mid west);
\end{tikzpicture}
\end{document}


And result:

Edit (in response to the comment)

To get the dashed arrows, we simply pull those ones out of the "Horizontal arrows" list and create a new loop for these, adding in the dashed option. Full code:

\documentclass{article}
%\url{http://tex.stackexchange.com/q/31687/86}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{%
matrix,%
calc,%
arrows%
}

\DeclareMathOperator{\coker}{coker}

\begin{document}
\begin{tikzpicture}[>=angle 60]
\matrix[matrix of math nodes,column sep={30pt},row sep={40pt,between origins}] (s)
{
&|[name=0a]| 0 &|[name=0b]| 0 &|[name=0c]| 0 \\
%
|[name=0k]| 0 &|[name=ka]| \ker \lambda &|[name=kb]| \ker \mu &|[name=kc]| \ker \nu \\
%
|[name=0A]| 0 &|[name=A]| L_1 &|[name=B]| M_1 &|[name=C]| N_1 &|[name=01]| 0 \\
%
|[name=02]| 0 &|[name=A']| L_0 &|[name=B']| M_0 &|[name=C']| N_0 &|[name=0C']| 0 \\
%
&|[name=ca]| \coker \lambda &|[name=cb]| \coker \mu &|[name=cc]| \coker \nu &|[name=0ck]| 0 \\
%
&|[name=0ca]| 0 &|[name=0cb]| 0 &|[name=0cc]| 0 \\
%
};

% Horizontal arrows
\foreach \start/\end in {%
A/B,
B/C,
C/01,
02/A',
A'/B',
B'/C',
ka/kb,
kb/kc,
ca/cb,
cb/cc%
} {
\draw[->] (\start.mid east) -- (\end.mid west);
}

% Horizontal dashed arrows
\foreach \start/\end in {%
0k/ka,
0A/A,
C'/0C',
cc/0ck%
} {
\draw[dashed,->] (\start.mid east) -- (\end.mid west);
}

% Vertical arrows
\foreach \start/\end in {%
0a/ka,
0b/kb,
0c/kc,
ka/A,
kb/B,
kc/C,
A/A',
B/B',
C/C',
A'/ca,
B'/cb,
C'/cc,
ca/0ca,
cb/0cb,
cc/0cc%
} {
\draw[->] (\start) -- (\end);
}

\path (A.mid east) -- node[auto] {$$\alpha_1$$} (B.mid west);
\path (B.mid east) -- node[auto] {$$\beta_1$$} (C.mid west);
\path (A'.mid east) -- node[auto,swap] {$$\alpha_0$$} (B'.mid west);
\path (B'.mid east) -- node[auto,swap] {$$\beta_0$$} (C'.mid west);
\path (A) -- node[auto,swap] {$$\lambda$$} (A');
\path (B) -- node[auto,swap,pos=.3] {$$\mu$$} (B');
\path (C) -- node[auto,swap] {$$\nu$$} (C');

\path (C.mid east) -- node (gap) {} (A'.mid west);

\draw[->] (kc.mid east) to[out=0,in=90] ($(kc)+(1,-.5)$) -- node[auto] {$$\delta$$} ($(C)+(1,.5)$) to[out=-90,in=0] (C.mid east) -- (gap) -- (A'.mid west) to[out=180,in=90] ($(A')+(-1.3,-.5)$) -- ($(ca)+(-1.3,.5)$) to[out=-90,in=180] (ca.mid west);
\end{tikzpicture}
\end{document}


And result:

-
Beautiful! And I can compile it without problems, too. May I ask of you to modify a little detail? Could you make the $0\to ker \lambda$, $0\to L_1$, $N_0\to 0$, $coker \nu \to 0$ dashed arrows? As you know, they are not really compulsory in the snake lemma ($\alpha_1$ monic implies $0\to ker \lambda$; $\beta_0$ epic implies $\coker \nu \to 0$). Thanks! (Sorry, but I'm not at all familiar with tikz...) – lentic catachresis Oct 17 '11 at 12:50
@BrunoStonek: Dashed as (I hope!) requested. – Loop Space Oct 17 '11 at 18:18

If you cheat a little bit you can get the following figure.

I replaced @cmhuhes's following lines

\nccurve[angleA=0,angleB=180]{->}{kern}{cokerl}
% npos takes a value between 0 and 1 for \nccurve
\naput[npos=0.1]{$\delta$}


with

\pnode(3.5,2.8){N11}
\pnode(0.5,2.2){N12}
\nccurve[angleA=0, angleB=10]{-}{kern}{N11}
\naput[npos=0.5]{$\delta$}
\psline(3.57,2.8)(0.45,2.15)
\nccurve[angleA=-150, angleB=-180]{->}{N12}{cokerl}

-