\documentclass{amsart}
\renewcommand{\datename}{Last updated:}


%%%%%%%%%%% Math  %%%%%%%%%%%%%%
\RequirePackage{amsthm}
\RequirePackage[alphabetic,initials]{amsrefs}
\RequirePackage{array}
\RequirePackage{enumerate}
\RequirePackage{cancel}
\RequirePackage{bbm}

%%%%%%%%%%% Graphics  %%%%%%%%%%%%%%
\RequirePackage{tikz}
\usetikzlibrary{matrix,arrows,backgrounds,shapes.misc,shapes.geometric}


\newcommand{\dimV}[1]{\begin{smallmatrix}#1\end{smallmatrix}}

%%
% Dimension vector commands
%%
\newcommand{\dimfv}[5]{\begin{smallmatrix}
						#1\\ #2\\ #3\\ #4\\ #5
					   \end{smallmatrix}}

\newcommand{\dimfr}[4]{\begin{smallmatrix}
						#1\\ #2\\ #3\\ #4
					   \end{smallmatrix}}

\newcommand{\dimvv}[3]{\begin{smallmatrix}
						#1\\ #2\\ #3
					   \end{smallmatrix}}

\newcommand{\dimv}[2]{\begin{smallmatrix}
						#1\\ #2
					   \end{smallmatrix}}
\newcommand{\ds}{\ \phantom{1}}
\newcommand{\ms}[1]{\ #1\ }

% Tell tikz to create pre-complied images.
% These the images will be contained in the folder /diagrams/'name of tex file'/
% contained in the folder your tex document is in.
% Tou must create a diagrams folder with the subfolder 'name of tex file'
% before running this in order for the external pictures to be placed in the 
% proper subdirectory.
% If you need to force a diagram to be remade place the command
% \tikzset{external/remake next} 
% before the \begin{tikzpicure} you want to remake. See pg 339 for more info.
\newcommand{\diagramfolder}{../diagrams/more-ar-quivers/}
\usetikzlibrary{external}
\tikzexternalize[prefix=\diagramfolder] 

\begin{document}
\title{More A-R Quivers}
\author{Lucas David-Roesler}
\date{\today}

\maketitle

Let $Q$ be the quiver 

\[\begin{tikzpicture}
\foreach \x in {1,2,...,8}
	\node [name = \x] at (\x,0) {$\x$};
\node[name=9] at (8,1) {$9$};
\node[name=10] at (8,-1) {$10$};

\foreach \n/\m in {1/2,2/3,3/4,4/5,6/5,7/6,8/7,9/8,8/10}
	\draw[->] (\n) -- (\m);
\path[dashed] (1) edge[bend left] (3) (5) edge[bend left] (7) (9) edge[bend left] (10);
\end{tikzpicture}\]

Here is $\Gamma(\cmod kQ)$.

%\tikzset{external/remake next} 
\[\begin{tikzpicture}
[%background grid/.style={draw,step=1,color=black!10},show background grid,
%scale=1.25,
%rotate=-90,
%every node/.style={rotate=-90}
]
\foreach[count=\ti from 1] \x/\y/\z in 
	{0/0/5,  
	 1/1/\dimv45, 1/-1/\dimv65 ,
	 2/2/\dimvv345, 2/0/\dimv{4\ 6}{5},
	 3/3/\dimfr2345, 3/-1/4, 3/1/\dimvv{3\ds}{4\ 6}{5},
	 4/0/\dimv34, 4/2/\dimfr{2\ds}{3\ds}{4\ 6}{5},
	 5/-1/3, 5/1/\dimvv234,5/3/6,
	 6/0/\dimv23,6/4/\dimv76,6/6/10,
	 7/-1/2,7/3/7,7/5/\dimvv{\ms8}{7\ 10}{6\ds\ },
	 8/-2/\dimv12,8/4/\dimv{\ms8}{7\ 10}, 8/6/\dimvv876,
	 9/-1/1, 9/3/\dimv8{10}, 9/5/\dimv87,9/7/\dimfr9876,
	 10/4/8, 10/6/\dimvv987,
	 11/5/\dimv98, 12/4/9}
	\node[name=\ti] at (\x,\y) {$\z$};
\foreach \m/\n in 
	{1/2, 1/3,
	 3/5,2/5,2/4,
	 5/7,5/8,4/8,4/6,
	 7/9,8/9,8/10,6/10,
	 9/11,9/12,10/12,10/13,
	 11/14,12/14,13/15,
	 14/17, 15/18,15/19,16/19,
	 17/20,18/21,19/21,19/22,
	 20/23,21/24,21/25,22/25,22/26,
	 24/27,25/27,25/28,26/28,
	 27/29,28/29,29/30}
	\draw[->] (\m) -- (\n);
\end{tikzpicture}\]



Let $Q$ be the quiver 
% \tikzset{external/remake next} 
\[\begin{tikzpicture}
\foreach \x in {1,2,...,7}
	\node [name = \x] at (\x,0) {$\x$};
\foreach \x/\n in {5/9,6/8,7/10}
	\node [name = \n] at (\x,-1) {$\n$};
	
\foreach \n/\m in {1/2,2/3,3/4,4/5,6/5,7/6,8/6,9/8,8/10}
	\draw[->] (\n) -- (\m);
\path[dashed] (1) edge[bend left] (3) (5) edge[bend left] (7) (10) edge[bend left] (9);
\end{tikzpicture}\]

Here is $\Gamma(\cmod kQ)$.

% \tikzset{external/remake next} 
\[\begin{tikzpicture}
[%background grid/.style={draw,step=1,color=black!10},show background grid,
%scale=1.25,
%rotate=-90,
%every node/.style={rotate=-90}
]
\foreach[count=\ti from 1] \x/\y/\z in 
	{0/0/5,  
	 1/1/\dimv45, 1/-1/\dimv65 ,1/-3/10,
	 2/-2/\dimvv{\ms8}{6\ 10}{5\ds \ },2/2/\dimvv345, 2/0/\dimv{4\ 6}{5},
	 3/-3/\dimvv865,3/-1/\dimvv{\ds 8}{4\ 6\ 10}{5\ \ds}, 3/1/\dimvv{3\ds}{4\ 6}{5}, 3/3/\dimfr2345,
	 4/-4/\dimfr9865,4/-2/\dimvv{\ds 8}{4\ 6}{5},4/0/\dimvv{3\ds 8\ds}{4\ 6\ 10}{5\ \ds}, 4/2/\dimfr{2\ds}{3\ds}{4\ 6}{5},
	 5/-3/\dimfr{\ds9}{\ds8}{4\ 6}{5} ,5/-1/\dimvv{3\ 8}{4\ 6}{5}, 5/1/\dimfr{2\ds\ds}{3\ds 8\ }{\ 4\ 6\ 10}{5\ds}, 5/3/6,
	 6/-4/4,6/-2/\dimfr{\ds 9}{3\ 8}{4\ 6}{5},6/0/\dimfr{2\ds}{3\ 8 }{4\ 6}{5}, 6/2/\dimv{8}{6\ 10},6/4/\dimv76 ,
	 7/-3/\dimv34,7/-1/\dimfr{2\ 9}{3\ 8}{4\ 6}{5}, 7/1/\dimv86,7/3/\dimv{7\ 8\ }{\ds 6\ 10},
	 8/-4/3, 8/-2/\dimvv234,8/0/\dimvv986,8/2/\dimv{7\ 8}{6},8/4/\dimv8{10},
	 9/-3/\dimv23,9/1/\dimvv{\ds 9}{7\ 8}{6}, 9/3/8 ,
	 10/-4/2,10/0/7, 10/2/\dimv98,
	 11/-5/\dimv12,11/3/9,
	 12/-4/1}
	\node[name=\ti] at (\x,\y) {$\z$};
\foreach \n/\m in 
	{1/2,1/3,
	  4/5,3/5,3/7,2/7,2/6,
	  5/8,5/9,7/9,7/10,6/10,6/11,
	  8/12,8/13,9/13,9/14,10/14, 10/15,11/15,
	  12/16,13/16,13/17,14/17,14/18,15/18,15/19,
	  16/20,16/21,17/21,17/22,18/22,18/23,19/23,19/24,
	  20/25,21/25,21/26,22/26,22/27,23/27,23/28,24/28,
	  25/29,25/30,26/30,26/31,27/31,27/32,28/32,28/33,
	  29/34,30/34,32/35,32/35,32/36,33/36,
	  34/37,35/38,35/39,36/39,
	  37/40,39/41,
	  40/42}
	\draw[->] (\n) -- (\m);
\end{tikzpicture}\]


Let $Q$ be the quiver

% \tikzset{external/remake next} 
\[\begin{tikzpicture}
\node[name=1] at (1,0) {1};
\node[name=2] at (2,0) {2};
\node[name=4] at (3,0) {4};
\node[name=3] at (4,0) {3};
\node[name=5] at (3,-1) {5};
\node[name=7] at (4,-1) {7};
\node[name=6] at (5,-1) {6};
\foreach \n/\m in {2/1,4/2,3/4,5/4,5/7,7/6}
	\draw[->] (\n) -- (\m);
\path[dashed] (2) edge[bend left] (3);
\path[dashed] (5) edge[bend left] (6);
\end{tikzpicture}\]
Here is $\Gamma(\cmod kQ)$.

 \tikzset{external/remake next} 
\[\begin{tikzpicture}
[%background grid/.style={draw,step=1,color=black!10},show background grid,
%scale=1.25,
%rotate=-90,
%every node/.style={rotate=-90}
]
\foreach[count=\ti from 1] \x/\y/\z in 
	{0/0/1,0/4/6,
	  1/1/\dimv12,1/5/\dimv76,
	  2/0/2,2/2/\dimvv421,2/4/7,
	  3/1/\dimv42, 3/3/\dimfr{5}{4\ 9}{2\ds}{1\ds},
	  4/0/4,4/2/\dimvv{5}{4\ 7}{2\ds}, 4/4/\dimfr5421,
	  5/-1/\dimv34,5/1/\dimv{5}{4\ 7},5/3/\dimvv542,
	  6/0/\dimv{3\ 5}{\ 4\ 7} ,6/2/\dimv54,
	  7/-1/\dimv57,7/1/\dimv{3\ 5}{4},
	  8/0/5, 8/2/3 }
	\node[name=\ti] at (\x,\y) {$\z$};
\foreach \n/\m in
	  {1/3,2/4,
	   3/5,3/6,4/7,
	   5/8,6/8,6/9,7/9,
	   8/10,8/11,9/11,9/12,
	   10/13,10/14,11/14,11/15,12/15,
	   13/16,14/16,14/17,15/17,
	   16/18,16/19,17/19,
	   18/20,19/20,19/21}
	\draw[->] (\n) -- (\m);
 
\end{tikzpicture}\]	

\end{document}