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authorNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
committerNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
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+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ % %
+ % This is file t-manual.tex ( version 14.08.2006 ) %
+ % %
+ % Diagram macros for tangles and braided Hopf algebras %
+ % %
+ % (Yu. Bespalov, V. Lyubashenko) %
+ % %
+ % available from CTAN and http://www.math.ksu.edu/~lub/ %
+ % %
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\documentclass[11pt]{article}
+
+\usepackage{amsmath}
+\usepackage{t-angles}
+%\usepackage[emtex]{t-angles}
+%\usepackage{t-erratr}
+
+\setlength{\textwidth}{16 true cm}
+\setlength{\textheight}{22.5 true cm}
+\setlength{\topmargin}{-1.4 true cm}
+\setlength{\oddsidemargin}{0mm}
+
+\title{{\Huge\tt t-angles.sty }
+\footnote{t-angles.sty is available from {\tt
+http://www.math.ksu.edu/$\sim$lub/} or from CTAN}
+\\
+\mbox{\Large\bf(Diagram macros for tangles and braided Hopf algebras)}
+}
+\author{Yu. Bespalov \and V. Lyubashenko}
+\date{Version 14.08.2006}
+
+\def\subsec#1{\subsection*{#1}\addcontentsline{toc}{subsection}{#1}}
+
+\def\hhmode{{{\tt hh}-mode }}
+\def\Command#1{\hbox{\tt\string#1}}
+\def\command#1{\ \hbox{\tt\string#1} \ }
+\def\SHOW#1#2{\begin{array}{c}
+ \begin{tangle}#1\end{tangle}\\
+ \hbox{\tt\string#2}
+ \end{array}}
+\def\Show#1{\SHOW#1#1}
+\def\Showh#1{\SHOW{\hh#1}#1}
+
+
+\bibliographystyle{amsplain}
+
+\begin{document}
+
+\maketitle
+\tableofcontents
+
+\section{Introduction}
+
+\subsection*{Usage:}
+\addcontentsline{toc}{subsection}{Usage}
+$$
+\vcenter{\hbox{\command{\usepackage[emtex]\{t-angles\}}}
+ \hbox{(for \command{emtex} drivers, \tt dviwin,}
+ \hbox{ \tt dvips, yap )}}
+\qquad \hbox{or} \qquad
+\vcenter{\hbox{\command{\usepackage\{t-angles\}} $\equiv$}
+ \hbox{\command{\usepackage[TPIC]\{t-angles\}}}
+ \hbox{(for TPIC drivers such as \tt dviwin,}
+ \hbox{ {\tt xdvi, dvips, yap, dvipdfm, kdvi} )}}
+$$
+To use with \command{kluwer.cls} add the option \command{kluwer}:\\
+\command{\usepackage[emtex,kluwer]\{t-angles\}} or
+\command{\usepackage[kluwer]\{t-angles\}}.
+
+The main option \command{TPIC} is executed by default. It can be
+overwritten by the antagonistic option \command{emtex}. These two
+options give slightly different *.dvi output, when they are used with
+\LaTeX. The package works also with pdf\LaTeX. In this case both
+options produce identical *.pdf output. Actually, the third option
+\command{pdflatex} is executed in this case automatically. You should
+not type \command{\usepackage[pdflatex]\{t-angles\}} in your file
+unless you want to prohibit its use with \LaTeX. Another way to produce
+*.pdf file is to apply \command{dvipdfm} to the *.dvi output, obtained
+with the \command{TPIC} option.
+
+Under pdf\LaTeX{} the information about slanted lines is stored in a file
+*.emp and read on the following pass. Consequently, the changes made in
+a tangle diagram are not reflected immediately in the *.pdf output. You
+may need several ($\simeq2$) runs of pdf\LaTeX{} to see the final
+picture.
+
+
+\subsec{Acknowledgments}
+ An optional parameter for (co)actions is proposed by Bernhard Drabant.
+The file \command{t-angles.sty} contains parts of
+\command{emlines2.sty} by Georg Horn and Eberhard Mattes and parts of
+\command{eepic.sty} by Conrad Kwok. PDF implementation of em\TeX{}
+specials is due to Hans Hagen. We have incorporated his con\TeX t
+support macros `em\TeX{} specials to PDF conversion' from
+\command{supp-emp.tex} distributed with Te\TeX. These parts of the code
+are used in the three options: \command{emtex}, \command{TPIC} and
+\command{pdflatex} respectively. To understand them the reader is
+invited to read comments in the original works. In order to distinguish
+between ordinary \LaTeX{} and pdf\LaTeX{} modes, Heiko Oberdiek's
+package \command{ifpdf.sty} is loaded.
+
+
+\subsection*{Main features:}
+\addcontentsline{toc}{subsection}{Main features}
+\begin{itemize}
+\item
+The environments
+$$
+\vcenter{\hbox{\tt \{tangle\}}
+ \hbox{\tt \{tanglec\}}
+ \hbox{\tt \{tangler\}}
+ \hbox{\tt \{tangles\}}}
+\qquad
+\vcenter{\hbox{are arrays with}
+ \hbox{one or more}
+ \hbox{column style:}}
+\qquad
+\vcenter{\hbox{\tt \{array\}\{l\}}
+ \hbox{\tt \{array\}\{c\}}
+ \hbox{\tt \{array\}\{r\}}
+ \hbox{\tt \{array\}} }
+\quad
+\vcenter{\hbox{(left)}
+ \hbox{(centered)}
+ \hbox{(right)}
+ \hbox{(any)}}
+$$
+respectively. Likewise {\tt \{array\}}, the {\tt \{tangles\}}
+environment allows an optional argument {\tt t} or {\tt b} to align
+the upper base line or the bottom of the tangle with the exterior
+base line: \verb+\begin{tangles}[b]{l*3cr}+.
+\item
+ \command\unitlens is the global length parameter.
+ Default value is \command{10 pt} .
+\item
+ \command\hstretch and \command\vstretch are relative length
+ parameters, horizontal and vertical stretch:
+\begin{equation*}
+\begin{split}
+\command\unith = \command\hstretch \%\;\; \text{of}\; \command\unitlens \,,
+\\
+\command\unitv = \command\vstretch \%\;\; \text{of}\; \command\unitlens \,,
+\end{split}
+\end{equation*}
+ set to an integer number of percents before the beginning of a tangle.
+\newline
+ Default settings are
+\command{\hstretch}{\tt 100} and \command{\vstretch}{\tt 100}.
+ The commands \command\hstretch and \command\vstretch should be used only
+ outside of tangle environments (with an exception of embedded tangle
+ environments).
+\item
+The commands \command{\hstr \{<number>\}} ,
+\command{\vstr \{<number>\}} can be used inside tangle environments
+instead of \command{\hstretch <number>} , \command{\vstretch <number>}.
+They will act within their \LaTeX{} scope.
+\item
+The height of every row is \ {\tt 2}\command\unitv
+or \command\unitv if the command \command\hh (see below) is used;
+\newline
+the widths of standard fragments are \ {\tt 0, .5, 1, 2, 3} or {\tt 4}
+\command\unith
+\item
+The command \command\hh obeys to \LaTeX{} scope rules. The command
+\command\HH acts in the same way as \command\hh but put at the
+beginning of a row works for the whole row in the \ {\tt \{tangles\}}
+environment.
+\item
+The style understands the commands
+\command\thinlines and \command\thicklines .
+\item
+The command \command{\step[<number>]} is used to produce horizontal space
+\command\kern{\tt <number>}\command\unith and works in any mode
+(inside and outside of the \command{tangle} environment).
+\begin{eqnarray*}
+&\command\step = \command{\step[1]}
+\qquad
+&\command\Step = \command{\step[2]}
+\\
+&\command\hstep = \command{\step[.5]}
+\quad
+&\command\hhstep = \command{\step[-.5]}
+\end{eqnarray*}
+\item
+Vertical spacing before the next row is produced by
+\ $\backslash\backslash[\langle\hbox{\tt vertical\_space}\rangle]$ \
+with optional argument (like in standard \ {\tt \{array\}} \
+environment).
+\item
+The command \command{\object\#1} is used to put the object \command{#1}
+directly over or under the end of the string (inside and outside of the
+\ {\tt \{tangle\}} \ environment). It adds a vertical space below or
+above as required. More space can be added as above.
+\newline
+The command
+\command{\Put(x\_coord,y\_coord)[binding\_point]<object>} puts
+{\tt <object>} into the intended position and works in
+\ {\tt \{tangle[cs]\}} \ environment like a combination of \command\put
+and \command\makebox. Coordinates are integers, measured in
+{\tt .1}\command\unith, {\tt .1}\command\unitv units;
+\command{binding\_point} is a combination of two letters \ {\tt lcr} \
+and \ {\tt tcb} \ according to the usual \LaTeX{} rules.
+\newline
+The commands
+\command{\nodeu\#1,} \command{\noded\#1,}
+\command{\nodel\#1,} \command{\noder\#1,}
+\command{\noderu\#1,} \command{\noderd\#1,}
+\command{\nodelu\#1,} \command{\nodeld\#1}
+produce zero boxes and put \ {\tt \#1} \ into the corresponding
+position.
+\newline
+\begin{minipage}{0.4\textwidth}
+The picture
+\begin{equation*}
+\vstretch 200 \hstretch 200
+\begin{tangle}
+\nodeu.\noded.\nodel.\noder.\noderu.\noderd.\nodelu.\nodeld.
+\end{tangle}
+\end{equation*}
+is described by the lines:
+\end{minipage}
+\begin{minipage}{0.5\textwidth}
+\begin{verbatim}
+\vstretch 200 \hstretch 200
+\begin{tangle}
+\nodeu.\noded.\nodel.\noder.
+\noderu.\noderd.\nodelu.\nodeld.
+\end{tangle}
+\end{verbatim}
+\end{minipage}
+\end{itemize}
+
+\section{Macros in pictures}
+
+\subsec{Straight lines and nodes}
+
+\begin{itemize}
+\item
+The commands \command\id, \command\n, \command\s,\command\node,
+\command\unit, \command\counit work also in \hhmode.
+$$
+\Show\id \quad
+\Show\idash \quad
+\Show\n \quad
+\Show\s \quad
+\Show\nd \quad
+\Show\sd \quad
+\Show\node \quad
+\Show\unit \quad
+\Show\counit
+$$
+The command \command\FillCircDiam denotes the filled circle diameter.
+It is set to an integer between 1 and 9 (here the measure unit is
+0.1 \command\unith). Default value is 3.
+\item
+The command \command\hln{\tt <number>} \ produces horizontal line on
+\ {\tt <number>}\command\unith.
+\item
+Argument of \command\ne,\command\nw,\command\se,\command\sw,
+\command\ned,\command\nwd,\command\sed,\command\swd
+is \ {\tt 0,1,2,3} or {\tt 4}; {\tt 0} produces empty box and
+other produce (dash) lines with horizontal projections equal
+\ {\tt <argument>}\command{\unith} cribbed into {\tt 1$\times$2} box.
+The commands \command\ne,\command\nw,\command\se,\command\sw
+produce {\tt 1$\times$1} box in \hhmode.
+$$
+\SHOW{\nw1}{\nw{1}}\quad
+\SHOW{\nwd1}{\nwd{1}}\quad
+\SHOW{\sw0}{\sw{0}}\quad
+\SHOW{\se3}{\se{3}}\quad
+\SHOW{\sed3}{\sed{3}}\quad
+\SHOW{\nw3\nw2\nw1\n\ne1\ne2\ne3}{\nw{3}\string\nw{2}%
+\string\nw{1}\string\n\string\ne{1}\string\ne{2}\string\ne{3}}
+$$
+\item
+All the following commands work in \hhmode and produce
+the similar diagrams in {\tt .5$\times$1} boxes.
+$$
+\Show\d \quad
+\Show\dd \quad
+\Show\hd \quad
+\Show\hdd \quad
+\Show\dh \quad
+\Show\ddh
+$$
+\end{itemize}
+
+
+\subsec{(Under/over)crossings. Braiding and symmetry}
+The following crossings and dashed crossings are shown in normal mode.
+$$
+\Show\x \quad
+\Show\xx \quad
+\Show\hx \quad
+\Show\hxx \quad
+\Show\X \quad
+\Show\XX \quad
+\Show\xd \quad
+\Show\xxd \quad
+\Show\hxd \quad
+\Show\hxxd
+$$
+The commands \command\X, \command\XX, \command\x, \command\xx work in
+\hhmode and produce similar diagrams of half width and height
+({\tt 1$\times$1} boxes).
+
+\subsec{(Co)pairings.}
+The commands \command\ev, \command\coev work in \hhmode and produce
+the similar diagrams of half width and height ({\tt .5$\times$1} boxes).
+For convenience in \hhmode \command\hev $\equiv$ \command\ev and
+\command\hev $\equiv$ \command\ev.
+\begin{align*}
+\Show\hev \quad
+&\Show\ev \quad
+&\Show\Ev \quad
+&\Show\EV
+&\\
+\Show\hcoev \quad
+&\Show\coev \quad
+&\Show\Coev \quad
+&\Show\COEV
+&\end{align*}
+
+
+\subsec{Morphisms. Frame and dash boxes}
+\begin{itemize}
+\item
+$$
+\SHOW{\Q f}{\Q \ f} \quad
+\SHOW{\QQ f}{\QQ \ f} \quad
+\SHOW{\O f}{\O \ f} \quad
+%\SHOW{\circ f}{\circ \ f} \quad
+\SHOW{\morph f}{\morph \ f} \quad
+\SHOW{\tu f}{\tu \ f} \quad
+\SHOW{\td f}{\td \ f} \quad
+\Show\S \quad
+\Show\SS
+$$
+\item
+$$
+\SHOW{\ox f}{\ox \ f} \qquad\qquad
+\SHOW{\ro p}{\ro \ p} \quad
+\SHOW{\coro q}{\coro \ q} \quad
+\SHOW{\Ro p}{\Ro \ p} \quad
+\SHOW{\coRo q}{\coRo \ q}
+$$
+\item
+The commands \command{\dbox\#1\#2,}\command{\ffbox\#1\#2,}
+\command{\obox\#1\#2,}\command{\tbox\#1\#2} put {\tt \$\#2\$} in the middle
+of \ {\tt \#1}$\times${\tt 2} (or {\tt \#1}$\times${\tt 1} \ in
+\hhmode ) box with dash, rectangle, oval frame or without frame.
+\par
+For example, the text
+\newline
+\begin{minipage}{.4\textwidth}
+\begin{verbatim}
+\begin{tangles}{rcl}
+\HH\obox 10&&\obox 10\\
+\HH\d&&\dd\\
+&\hhstep\obox 3V\hhstep&
+\end{tangles}
+\end{verbatim}
+\end{minipage}
+produces
+\begin{minipage}{.4\textwidth}
+\begin{equation*}
+\begin{tangles}{rcl}
+\HH\obox 10&&\obox 10\\
+\HH\d&&\dd\\
+&\hhstep\obox 3V\hhstep&
+\end{tangles}
+\end{equation*}
+\end{minipage}
+\end{itemize}
+
+\subsec{(Co)multiplications and cocycles}
+The commands \command\cu, {\tt \string\cu *},
+\command\cd, {\tt \string\cd *} work in \hhmode and produce the similar
+diagrams of half width and height ({\tt .5$\times$1} boxes).
+For convenience in \hhmode \command\hcu $\equiv$ \command\cu and
+\command\hcd $\equiv$ \command\cd .
+\begin{equation*}
+\begin{split}
+\Show\Cu \quad
+\Show\cu \quad
+\Show\hcu \quad
+\Show\hdcu \quad
+\Show\hddcu
+\\
+\SHOW{\Cu *}{\Cu *} \quad
+\SHOW{\cu *}{\cu *} \quad
+\SHOW{\hcu *}{\hcu *} \quad
+\SHOW{\hdcu *}{\hdcu *} \quad
+\SHOW{\hddcu *}{\hddcu *}
+\\
+\SHOW{\Cd *}{\Cd *} \quad
+\SHOW{\cd *}{\cd *} \quad
+\SHOW{\hcd *}{\hcd *} \quad
+\SHOW{\hdcd *}{\hdcd *} \quad
+\SHOW{\hddcd *}{\hddcd *}
+\\
+\Show\Cd \quad
+\Show\cd \quad
+\Show\hcd \quad
+\Show\hdcd \quad
+\Show\hddcd
+\end{split}
+\end{equation*}
+
+\subsec{(Co)actions}
+Commands \command\lu, \command\ld, \command\ru, \command\rd
+have optional parameter \ {\tt [\#1]} which equals to width of the box:
+\begin{align*}
+&\command\hh
+& \SHOW{\hh\Lu}{\Lu$\equiv\!\!\!\!$\command\lu$\!\!\!\!\!\!$[2]} \quad
+ \SHOW{\hh\lu} {\lu$\equiv\!\!\!\!$\command\lu$\!\!\!\!\!\!$[1]} \quad
+ \Showh\hlu \quad
+ \Showh\hru \quad
+ \SHOW{\hh\ru} {\ru$\equiv\!\!\!\!$\command\ru$\!\!\!\!\!\!$[1]} \quad
+ \SHOW{\hh\Ru} {\Ru$\equiv\!\!\!\!$\command\ru$\!\!\!\!\!\!$[2]}
+&\\
+&&\SHOW\Lu{\Lu$\equiv\!\!\!\!$\command\lu$\!\!\!\!\!\!$[2]} \quad
+ \SHOW\lu{\lu$\equiv\!\!\!\!$\command\lu$\!\!\!\!\!\!$[1]} \quad
+ \Show\hlu \quad
+ \Show\hru \quad
+ \SHOW\ru{\ru$\equiv\!\!\!\!$\command\ru$\!\!\!\!\!\!$[1]} \quad
+ \SHOW\Ru{\Ru$\equiv\!\!\!\!$\command\ru$\!\!\!\!\!\!$[2]}
+&\\
+&&\SHOW\Ld{\Ld$\equiv\!\!\!\!$\command\ld$\!\!\!\!\!\!$[2]} \quad
+ \SHOW\ld{\ld$\equiv\!\!\!\!$\command\ld$\!\!\!\!\!\!$[1]} \quad
+ \Show\hld \quad
+ \Show\hrd \quad
+ \SHOW\rd{\rd$\equiv\!\!\!\!$\command\rd$\!\!\!\!\!\!$[1]} \quad
+ \SHOW\Rd{\Rd$\equiv\!\!\!\!$\command\rd$\!\!\!\!\!\!$[2]}
+&\\
+&\command\hh
+& \SHOW{\hh\Ld}{\Ld$\equiv\!\!\!\!$\command\ld$\!\!\!\!\!\!$[2]} \quad
+ \SHOW{\hh\ld}{\ld$\equiv\!\!\!\!$\command\ld$\!\!\!\!\!\!$[1]} \quad
+ \Showh\hld \quad
+ \Showh\hrd \quad
+ \SHOW{\hh\rd}{\rd$\equiv\!\!\!\!$\command\rd$\!\!\!\!\!\!$[1]} \quad
+ \SHOW{\hh\Rd}{\Rd$\equiv\!\!\!\!$\command\rd$\!\!\!\!\!\!$[2]}
+&\end{align*}
+
+\subsec{Compositions}
+$$
+\Show\kk \quad
+\Show\luld \quad
+\Show\cucd \quad
+\Show\rurd \quad
+\Show\k
+$$
+
+\section{Examples}
+
+\begin{minipage}{.3\textwidth}
+\[
+\begin{tangle}
+\object{B}\step\object{B}\\
+\cucd\\
+\object{B}\step\object{B}
+\end{tangle}
+\;=\enspace
+\begin{tangles}{lcr}
+\HH \cd && \cd \\
+\HH \id & \x & \id \\
+\HH \cu && \cu
+\end{tangles}
+\]
+\begin{equation*}
+\hstretch 90 \vstretch 60
+\begin{tangle}
+\step\hcoev\step\coev\\
+\dd\step\hxx\step\dd\\
+\id\Step\id\step\hx\\
+\d\step\hxx\step\d\\
+\step\hev\step\ev
+\end{tangle}
+\end{equation*}
+\end{minipage}
+\begin{minipage}{.2\textwidth}
+The first picture is produced by:
+\end{minipage}
+\quad
+\begin{minipage}{.4\textwidth}
+\begin{verbatim}
+\[
+\begin{tangle}
+\object{B}\step\object{B}\\
+\cucd\\
+\object{B}\step\object{B}
+\end{tangle}
+\;=\enspace
+\begin{tangles}{lcr}
+\HH \cd && \cd \\
+\HH \id & \x & \id \\
+\HH \cu && \cu
+\end{tangles}
+\]
+\end{verbatim}
+\end{minipage}
+
+\begin{minipage}{0.23\textwidth}
+Nested environments:
+\[
+\begin{tanglec}
+ \begin{tangles}[b]{c}
+ \vstr{200} \xx \\ \vstr{200} \xx
+ \end{tangles}
+\step[4]
+ \begin{tangles}[b]{*3c}
+ & \object{F} & \\ & \xx & \\ \id && \O{u_1^2} \\
+ & \xx & \\ \id && \O{u_1^2}
+ \end{tangles}
+\\
+\id \Step \coRo\omega \Step \id \\
+\hstr{200} \coRo\omega
+\end{tanglec}
+\]
+The level of nesting depends on the save size of your \TeX.
+\end{minipage}
+\ \
+\begin{minipage}{0.7\textwidth}
+\begin{verbatim}
+\[
+\begin{tanglec}
+ \begin{tangles}[b]{c}
+ \vstr{200} \xx \\ \vstr{200} \xx
+ \end{tangles}
+\step[4]
+ \begin{tangles}[b]{*3c}
+ & \object{F} & \\ & \xx & \\ \id && \O{u_1^2} \\
+ & \xx & \\ \id && \O{u_1^2}
+ \end{tangles}
+\\
+\id \Step \coRo\omega \Step \id \\
+\hstr{200} \coRo\omega
+\end{tanglec}
+\]
+\end{verbatim}
+\end{minipage}
+\smallskip
+
+Note the use of optional argument \verb+[b]+ to align the subtangles
+at the bottom.
+
+\section{Development}
+
+\subsec{History and versions}
+
+The style was produced by the first author in 1994. It was completely
+modified and essentially improved by the second author in 1997 for
+real--life applications in \cite{BKLT:int}.
+
+{%beginenumerate
+\setlength{\leftmargini}{34mm}
+\begin{enumerate}
+\item[04.04.99$\to$20.04.00]
+The output of commands \command{\tu\#1,}\command{\td\#1,}
+\command{\ro\#1,}\command{\coro\#1,}\command{\Ro\#1,}\command{\coRo\#1}
+slightly differs. Now they fit their boxes.
+
+\item[20.04.00$\to$10.09.00]
+Dashed crossings are represented by the commands
+\command{\xd,}\command{\xxd,}\command{\hxd,}\command{\hxxd.}
+
+\item[10.09.00$\to$22.04.06]
+It is possible to use the package with pdf\LaTeX.
+
+\item[22.04.06$\to$14.08.06]
+Behaviour of the package with pdfe\LaTeX{} of MiK\TeX{} 2.5 is
+corrected.
+
+\end{enumerate}
+}%endenumerate
+
+\begin{thebibliography}{1}
+
+\bibitem{BKLT:int}
+Yu.~N. Bespalov, T.~Kerler, V.~V. Lyubashenko, and V.~G. Turaev,
+ \emph{Integrals for braided {H}opf algebras}, J. Pure and Appl. Algebra
+ \textbf{148} (2000), no.~2, 113--164, Available as
+ http://arXiv.org/abs/q-alg/9709020.
+
+\end{thebibliography}
+
+%\bibliography{yuri}
+
+%\subsec{Bugs}
+%\begin{itemize}
+%\item
+%\end{itemize}
+
+\subsec{Directions for modification
+%\protect\footnote{What can you suggest?}
+}
+
+\begin{itemize}
+\item
+In the future some problems can be solved by introducing global
+(logical) parameters that switch configuration and behavior of certain
+families of commands in questionable situations.
+\item
+To adopt commands like in {\tt \{picture\}} environment to
+produce special fragments of one time use.
+\item
+To make the second argument of the command \command{\Put(\#1)[\#2]\#3}
+optional.
+\item
+To produce command index for this manual.
+\item
+To add possibility to change size of circle in circled morphisms
+(in particular, to turn \command\morph into a special case of \command\O).
+\end{itemize}
+
+
+\medskip
+\begin{center}
+\fbox{Suggestions are welcome.}
+\end{center}
+
+
+\appendix
+\section{Exercises}
+How to produce the following ?
+\begin{equation*}
+\vstretch 300
+\begin{tanglec}
+\hh\dd\d\step[1.5]\dd\d \\
+\obox 6{\hstretch 85 \vstretch 80
+ \begin{tangles}{rcl}
+ \HH\obox 10&&\obox 10\\
+ \HH\d&&\dd\\
+ &\hhstep\obox 3V\hhstep&
+ \end{tangles} }
+\end{tanglec}
+\end{equation*}
+
+\end{document}
+
+
+\section{\tt t-erratr.sty}
+
+Macros from the old versions of \ {\tt tangles.sty} \ are stored in
+this file for compatibility.
+\begin{itemize}
+\item
+The following commands are special cases of
+\command{\ffbox\#1\#2,}\command{\dbox\#1\#2}:
+\begin{align*}
+&\SHOW{\dash f}{\dash \ f} \enspace
+&\SHOW{\Dash f}{\Dash \ f} \enspace
+&\SHOW{\DDash f}{\DDash \ f} \quad
+&\SHOW{\Frabox f}{\Frabox \ f} \enspace
+&\SHOW{\doubleFrabox f}{\doubleFrabox \ f}
+\\
+\command\hh
+&\SHOW{\hh\dash f}{\dash \ f} \enspace
+&\SHOW{\hh\Dash f}{\Dash \ f} \enspace
+&\SHOW{\hh\DDash f}{\DDash \ f} \quad
+&\SHOW{\hh\Frabox f}{\Frabox \ f} \enspace
+&\SHOW{\hh\doubleFrabox f}{\doubleFrabox \ f}
+\end{align*}
+and \command{\frabox} is equivalent to \command{\Frabox}.
+\item
+The following morphisms are special cases of
+\command{\ro\#1,} \command{\coro\#1,} \command{\Ro\#1,} \command{\coro\#1}:
+\vstretch 150 \hstretch 150
+\begin{align*}
+&\hstep\Show\r
+&\hstep\Show\rh
+&\hstep\Show\ra
+&\hstep\Show\rb
+\\
+&\hstep\Show\rr
+&\hstep\Show\rrr
+&\hstep\Show\rra
+&\hstep\Show\rrb
+\\
+&\Show\R
+&\Show\Rh
+&\Show\Ra
+&\Show\Rb
+\\
+&\Show\RR
+&&\Show\RRa
+&\Show\RRb
+\end{align*}
+\item
+The following commands are special cases of \command{\O\#1}:
+$$
+\vstretch 150 \hstretch 150
+\Show\tS \quad
+\Show\tSS \quad
+\SHOW{\Ointl H}{\Ointl \ H} \;
+\SHOW{\Ointr H}{\Ointr \ H} \;
+\SHOW{\Ocointl H}{\Ocointl \ H} \;
+\SHOW{\Ocointr H}{\Ocointr \ H}
+$$
+\item
+$$
+\SHOW{\pairing p}{\pairing \ p} \quad
+\SHOW{\Pairing P}{\Pairing \ P} \quad
+$$
+\item
+Functions of the old unsightly commands
+\command{\obj\#1,} \command{\Obj\#1,} \command{\negobj\#1}
+are shouldered by the command \command{\Put(\#1)[\#2]\#3}.
+\end{itemize}
+
+Commands \command{\harc}, \command{\inarc}, \command{\outarc} have two
+arguments -- horizontal and vertical diameters of an ellipse
+\vspace*{5mm}
+\[
+\SHOW{\inarc{15}{20}}{\inarc \{15\}\{20\}} \qquad
+\SHOW{\harc{20}{15}}{\harc \{20\}\{15\}} \qquad
+\SHOW{\outarc{30}{25}}{\outarc \{30\}\{25\}}
+\]
+
+\end{document}