\newcommand\myemph\emph

\newcommand\notch{^{(p)}}
\newcommand \tilblacktri[1]{\widetilde{\triangle}_{#1}}
\newcommand \blacktri[1]{\triangle_{#1}}
\newcommand \tiltaui[1] {\widetilde{\tau}_{i_{#1}}}
\newcommand \taui[1] {\tau_{i_{#1}}}


\newcommand \al {\alpha} \newcommand \be {\beta}
\newcommand\ga{\gamma}  
\newcommand\Ga{\Gamma}
\newcommand\ta{\tau } \newcommand\si{\sigma}  \newcommand\Si{\Sigma}
\newcommand \om {\omega} \newcommand \oom {$w$}
\newcommand\Cn{\mathcal{C}_n}
\newcommand\Surf{S}
\newcommand\Cnn[1]{\mathcal{C}_{#1}}
\newcommand\Z{\mathbb{Z}}
\newcommand\B{\mathcal B}
\newcommand\AAA {\mathcal A}

\newcommand\tgk{\tau_{[\ga_k]}}
\newcommand\tbk{\tau_{[\be_k]}}
\newcommand\tgl{\tau_{[\ga_\ell]}}
\newcommand \llrrl {\ell, \ell, r, r, \ell} \newcommand \lrrll {\ell, r, r, \ell, \ell}
\newcommand\rrBak{\underline{r,r}}
\newcommand\rrNon{\mathbf{r,r}}
\newcommand\ttBak{\underline{\tau, \tau}}
\newcommand\ttNon{\boldsymbol{\tau, \tau}}
\newcommand\ttQuasi{\overline{\tau, \tau}}
\newcommand\ttQuasiNon{\overline{\boldsymbol{\tau},\boldsymbol{\tau}}}
\newcommand\dotRR{\dot{\textbf{r}}, \textbf{r}}
\newcommand\RddotR{\textbf{r}, \ddot{\textbf{r}}}
\newcommand\GT {G_{T^o,\ga}}
\newcommand\overGT {\overline{G}_{T^o,\ga}}
\newcommand\puncture{\text{\tiny $P$}}
\newcommand\Quad[1]{{\rm Quad}{(#1)}}

\newcommand \tilp {\widetilde{p}} \newcommand \tilP {\widetilde{P}}
\newcommand\tilv{\widetilde{v}} \newcommand\tily{\widetilde{y}} \newcommand\tilz{\widetilde{z}}
\newcommand\tilt{\tilde{t}} \newcommand\tils{\tilde{s}} \newcommand\tilr{\widetilde{r}} \newcommand\tilell {\widetilde{\ell}}
\newcommand\tilS{\widetilde{\Sga}}  \newcommand\Sga{S_\ga}
\newcommand\tilA{\widetilde{A}} \newcommand\tilD{\widetilde{D}}
\newcommand\tilga{\widetilde{\ga}}
\newcommand\tilsi{\widetilde{\si}}
\newcommand\tilw{\widetilde{\om}}
\newcommand\tilal{\widetilde{\al}}
\newcommand\tilbe{\widetilde{\be}}
\newcommand\tillam{\widetilde{\lambda}}
\newcommand\tiltau{\widetilde{\tau}}
 \newcommand\tilom{\widetilde{\om}}
\newcommand\overpi{\overline{\pi}}
\newcommand\tilT{\widetilde{T}}
\newcommand\tilTo{\widetilde{T^o}}
\def\a{\aleph}
\def\b{\text{\tiny $\beth$}}
\newcommand\ccr{r^{cc}}
\newcommand\bijection{\longleftrightarrow}

\newcommand\ts[1]{\tau_{[\si_{#1}]}}
\newcommand\tg[1]{\tau_{[\ga_{#1}]}}
\newcommand\tb[1]{\tau_{[\be_{#1}]}}
\newcommand\tiltg[1]{\widetilde{\tau}_{[\ga_{#1}]}}
\newcommand \pkpkone[1] {[p_k,p_{k+1}]_{#1}}
\newcommand\pipjOm[2]{[p_{#1}, \dots, p_{#2}]_{\om}}
\newcommand\overom{\bar{\om}}
%\newcommand\disk[1]{\textbf{Disk}_{#1}}
\newcommand\MathDisk[1]{\textbf{Disk}_{#1}}
\newcommand\tildisk[1]{\textbf{Disk}_{\widetilde{#1}}}
\newcommand\tildiskplus[2]{\textbf{Disk}_{\widetilde{#1}}^{#2}}
\newcommand\outsidedisk[1]{\textbf{C}_{#1}}

\newcommand\PMOne{(b_4, 	 	b_2, 	 	b_3, 		 		r, 	\ell, 	b_3)}

\newcommand\PMTwo{(b_4, 	1, 	\ell, 	r, 	\ell, 	b_3)}

\newcommand\TpathThree{(b_1, 1, 2, \ell, 	2)}
\newcommand\PMThree{(b_1,	2,2,\ell, r, 2)}

\newcommand\PMFour{(b_1, 2, b_3, r,\ell, b_3)}

\newcommand\PMFive{(b_4, b_2, 2, \ell,r,2)}

\newcommand\PMSix{(b_4, 	b_2,2,	\ell,	\ddot{r},	\ell, 	b_3)}

\newcommand\PMSeven{(b_4, b_2, 2, \dot{r}, \ell, b_3)}

\newcommand\PMEight{(b_1, 2,2, \ell,	\ddot{r}, b_3)}

\newcommand\PMNine{(b_1, 2, 2, \dot{r}, \ell, b_3)}



%% Next two commands enlarge space for figures on the top of a page
\renewcommand{\textfraction}{0.01}
\renewcommand{\topfraction}{0.99}


\newcommand{\zg}{\gamma} % used
\newcommand{\zD}{\Delta} % used

\newcommand{\Brac}{\text{Brac}} % used
\newcommand{\Cheb}{\text{Cheb}} % used
%\newcommand{\cross}{\text{cross}} % used
\newcommand{\F}{\mathcal{F}} % used
\newcommand{\N}{\mathbb{N}} % used

%\newtheorem*{exam*}{Example}
%\newtheorem{application}[theorem]{Application}

\newcommand\notchP{^{(\text{\tiny $P$})}}

\newcommand\Om{\Omega}

\newcommand\Tganotch{(T^{\circ},\ga\notchP)}
\newcommand\TganotchP{(T^{\circ},\ga\notchP)}
\newcommand\TganotchPQ{(T^{\circ},\ga\notchPQ)}
\newcommand\overx{\overline{x}}
\newcommand\overy{\overline{y}}
\newcommand\overoverx{\overline{\overline{x}}}
\newcommand\overovery{\overline{\overline{y}}}
\newcommand\tile{\square}

\newcommand\punctureP{\text{\tiny{P}}}

\newcommand{\ellorientation}{{clockwise}}
\newcommand{\ellOpOrientation}{counterclockwise}
\newcommand\disk{{disk}}
\newcommand\disks{{disks}}

\newcommand\cross{\text{cross}}
\newcommand\bargammanotchP{\overline{\ga}\notchP}
\newcommand\bargammanotchPQ{\overline{\ga}\notchPQ}

\newcommand\TextRestriction{{\text{restr}}}

\newcommand\signedadjacencymatrixofanidealtriangulation{signed adjacency matrix of an ideal triangulation}


%%%%