%% $Id: xyknot.doc,v 3.4 1997/05/18 01:13:24 ross Exp $
%%
%% Xy-pic ``Knots and Links'' feature.
%% Copyright (c) 1994-1996	Ross Moore	<ross@mpce.mq.edu.au>
%%
%% This file is part of the Xy-pic package for graphs and diagrams in TeX.
%% See the companion README and INSTALL files for further information.
%% Copyright (c) 1991-1996	Kristoffer H. Rose	<krisrose@brics.dk>
%%
%% The Xy-pic package is free software; you can redistribute it and/or modify
%% it under the terms of the GNU General Public License as published by the
%% Free Software Foundation; either version 2 of the License, or (at your
%% option) any later version.
%%
%% The Xy-pic package is distributed in the hope that it will be useful, but
%% WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
%% or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
%% for more details.
%%
%% You should have received a copy of the GNU General Public License along
%% with this macro package; if not, write to the Free Software Foundation,
%% Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
%%
\ifx\xyloaded\undefined \input xy \fi

\xyprovide{knot}{Knots and Links feature}{\stripRCS$Revision: 3.4 $}%
 {Ross Moore}{ross@mpce.mq.edu.au}%
 {Mathematics Department, Macquarie University, NSW~2109, Australia}

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent
 This feature provides a language for specifying knots, links and  
 general arrangements of crossing strings. 

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph*{Header:}\leavevmode
\DOCHEADER

\TODO: Document this feature!

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\DOCMODE(
\message{knots and links,}
\DOCMODE)

This |knot| feature is really a `construction kit', providing pieces which 
may be placed appropriately to form knots and links. 
The types of pieces provided are of two kinds: the ``crossings'', 
representing one string crossing over or under another; 
and ``joins'' which are used to connect what would otherwise be loose ends. 
Several types of each are provided, along with a simple way of specifying 
where to place arrowheads and labels.

\begin{figure*}[tb]
\begin{syntax}
%
 <knot-piece>
  &\iss &  <piece><scale><knot-labels> 
	& interpret knot-piece 
\cr
\noalign{\smallskip}
%
 <piece>
  &\iss & <crossing>\orr<join>
	& piece is a ??!^[crossing] or a ??!^[join] 
\cr
\noalign{\smallskip}
%
 <scale>
  &\iss & <empty>\orr~|-|~\orr |[|<num>|]| 
	& invert or ??!^[scale the knot piece]; \cr
& & \orr |~|<pos><pos><pos><pos>
  &  ??!^[alter size and shape] using the <pos>s 
\cr
%
\noalign{\smallskip}
<knot-labels> 
 &\iss & <empty>\orr <knot-tips><knot-labels> 
	 & arrowtips at ends, aligned with orientation
\cr
%
\noalign{\smallskip}
 & & \orr <where><what><knot-labels> 
	&  list??^[knot-labels] of ??!^[arrowtips, breaks and labels]
\cr
%
\noalign{\smallskip}
 & & \orr |@|<adjust><knot-labels> 
	&  ??!^[adjust hole] position for a <crossing>;
  \newline adjust other parameter??^[join control] for a <join>.
\cr
%
\noalign{\smallskip}
 <knot-tips>
 & \iss &~~~|==|\orr|=!| & ??!^[arrowtips] at both/neither end
 \cr 
 & & \orr|=<|\orr|=>| & ??!^[arrowtips] also at start/finish 
\cr
\noalign{\smallskip}
%
 <where>
 & \iss  &~~~||||\orr||||<adjust> & `over' string on a <crossing>;%
 ??^[which string] 
 \newline middle??^[join places] place on a <join>.
\cr
  & & \orr|<|\orr|<|<adjust> & initial portion of `under' string 
  on a <crossing>;??^[which string] 
 \newline earlier??^[join places] place on a <join>.
\cr
 & & \orr|>|\orr|>|<adjust> & final portion of `under' string
  on a <crossing>;??^[which string] 
 \newline later??^[join places] place on a <join>.
\cr
%
\noalign{\smallskip}
 <adjust>
 & \iss &~~~|(+|<num>|)|\orr|(-|<num>|)| 
  & ??!^[adjustment] from current value of parameter
\cr
 & & \orr|(=|<num>|)|\orr|(|<num>|)| & set parameter value??^[adjustment] 
\cr
\noalign{\smallskip}
%
 <what>
  &\iss & ~~~|>|\orr|<| & arrowhead aligned with/against ??!^[orientation]
\cr
 & & \orr|\knothole|\orr|\khole| & leave ??!^[hole in the string] 
\cr
 & & \orr|{|<text>|}| & set??^[simple label] <text> as label, 
  using |\labelstyle| 
\cr
 & & \orr|{*|<object>|}| & drop <object>??^[knot object]
\cr
 & & \orr|{|<anchor><it>|}| & <break> or label??^[knot object] 
 as on an <arrow>
\cr
 & & \orr|||| & ??!^[null-break] 
\cr
%
\end{syntax}
\caption{\protect<knot-piece> construction set}
??=[f.path]
\end{figure*}


All the pieces ultimately use curves from the |curve| extension, usually
indirectly via the |arrow| feature. 
As such, processing can be memory-intensive and may seem rather slow. 
All the warnings and advice given elsewhere on techniques to handle pages
and individual diagrams with many curves are especially applicable when 
using this feature.

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Most constructions use |\ar| so make sure this feature is loaded.

\DOCMODE(
\xyrequire{arrow}\xycatcodes 
\DOCMODE)
\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\centerline{
%
\UseComputerModernTips
\vtop{\vbox{\xygraph{!{0;/r1.5pc/:} !{\vover}[u]
  !{\hcap[-2]<<} [d] !{\vover-} [ruu] !{\hcap[2]><}}\smallskip}
\hbox{\qquad\ \small simple link}}\hfil
%
% Figure-eight
%
\vtop{\vbox{\xygraph{!{0;/r1.5pc/:}
 [u] !{\vloop<(-.005)\khole||\vcrossneg \vunder- }
 [] !{\ar @{-}@'{p-(1,0)@+}+(-1,1)}
 [ul] !{\vcap[3]>\khole}
 [rrr] !{\ar @{-}@'{p-(0,1)@+}-(1,1)}
}\smallskip}\hbox{\small figure-8 knot}}\hfil\quad
%
% cinque-foil
%
\vtop{\vbox{\xygraph{!{0;/r1.25pc/:}
 !P5"a"{~>{}} !P10"b"{~:{(1.7487,0):}~>{}}
 !P20"c"{~={-9}~:{(2.2,0):}~>{}}
 !{\xunderv~{"b3"}{"b2"}{"a2"}{"a1"}}
 !{\xunderv~{"b5"}{"b4"}{"a3"}{"a2"}}
 !{\xunderv~{"b7"}{"b6"}{"a4"}{"a3"}}
 !{\xunderv~{"b9"}{"b8"}{"a5"}{"a4"}}
 !{\xunderv~{"b1"}{"b10"}{"a1"}{"a5"}}
 !{\vloop~{"c3"}{"c2"}{"b2"}{"b1"}<<}
 !{\vloop~{"c7"}{"c6"}{"b4"}{"b3"}<<}
 !{\vloop~{"c11"}{"c10"}{"b6"}{"b5"}<<}
 !{\vloop~{"c15"}{"c14"}{"b8"}{"b7"}<<}
 !{\vloop~{"c19"}{"c18"}{"b10"}{"b9"}<<}
}\smallskip}\hbox{\quad\ \small cinquefoil}}
}%

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Crossings}

A ``crossing'' is intended to represent two strings passing close by, 
but not meeting. The macros provided specify typesetting within a square
cell of coordinate values; using a non-square basis alters this shape,
but see also note~??[alter size and shape] below, for the technique
that was used in the ``cinquefoil'' example above.

\begin{notes}
\note??=[crossing]
Several families of crossing are provided. Those having names as |\v...| 
and |\h...| are designed to stack respectively vertically and
horizontally. More precisely the current <pos> starts at the top-left
corner and finishes at either the bottom-left or top-right. 
Say that a crossing is either a `vertical crossing' or `horizontal 
crossing' respectively.

This certainly applies to the |\..cross..| and |\..twist..| families, 
see figure~??[f.crossings] in which the strings enter and leave the 
square all with vertical tangents or all with horizontal tangents.
Indeed {\em all} crossings are either vertical or horizontal, with
the final letter indicating which for the |\xover..| families.

Furthermore there is a natural {\em orientation\/} for each crossing, 
as well as along each strand. This corresponds to the order in which ink 
is applied to the printed page, following the natural parametrization of 
each strand as a curved connection or arrow. 
This orientation determines whether a crossing is `over' (mathematically,
positive or right-handed) or `under' (mathematically, negative or 
left-handed). It is used in determining the location of labels and the 
direction of arrowheads placed along the strings. 
Note that |\..cross..| and |\..twist..| crossings may set the same
curves, but with different orientation and label-positioning.

\begin{figure*}[ht]
$$
\knotholesize{5pt}\UseComputerModernTips
 \xygraph{!{0;/r2pc/:}="O"
  []   !{\vcross<>|>>><{x}|{y}>{z}}    
  [rur]!{\vcrossneg<>|>>><{x}|{y}>{z}}
  [rur]!{\vuncross<>|>>><{x}|{y}>{z}} 
 [urrr]!{\hcross<>|>>><{x}|{y}>{z}}
    [r]!{\hcrossneg<>|>>><{x}|{y}>{z}}     
    [r]!{\huncross<>|>>><{x}|{y}>{z}}
 "O" !{+(.5,-1.5)}
  []  !{*\txt\small\tt{\string\vcross}} 
  [rr]!{*\txt\small\tt{\string\vcrossneg}} 
  [rr]!{*\txt\small\tt{\string\vuncross}}
 [rrr]!{*\txt\small\tt{\string\hcross}} 
  [rr]!{*\txt\small\tt{\string\hcrossneg}}
  [rr]!{*\txt\small\tt{\string\huncross}}
 "O" !{+(0,-3)}
 []   !{\vtwist<>|>>><{x}|{y}>{z}}    
 [rur]!{\vtwistneg<>|>>><{x}|{y}>{z}}
 [rur]!{\vuntwist<>|>>><{x}|{y}>{z}} 
 [urrr]!{\htwist<>|>>><{x}|{y}>{z}}
 [r]!{\htwistneg<>|>>><{x}|{y}>{z}}     
 [r]!{\huntwist<>|>>><{x}|{y}>{z}}
 "O" !{+(.5,-4.5)}
 !{*\txt\small\tt{\string\vtwist}} 
 [rr]!{*\txt\small\tt{\string\vtwistneg}} 
 [rr]!{*\txt\small\tt{\string\vuntwist}}
 [rrr]!{*\txt\small\tt{\string\htwist}} 
 [rr]!{*\txt\small\tt{\string\htwistneg}}
 [rr]!{*\txt\small\tt{\string\huntwist}}
 "O" !{+(0,-6)}
     []!{\xoverv<>|>>><{x}|{y}>{z}}
  [urr]!{\xunderv<>|>>><{x}|{y}>{z}} 
  [urr]!{\xunoverv<>|>>><{x}|{y}>{z}}
 [urrr]!{\xoverh<>|>>><{x}|{y}>{z}}
    [r]!{\xunderh<>|>>><{x}|{y}>{z}}
    [r]!{\xunoverh<>|>>><{x}|{y}>{z}}
 "O" !{+(.5,-7.5)}
      !{*\txt\small\tt{\string\xoverv}}
  [rr]!{*\txt\small\tt{\string\xunderv}} 
  [rr]!{*\txt\small\tt{\string\xunoverv}}
 [rrr]!{*\txt\small\tt{\string\xoverh}}
  [rr]!{*\txt\small\tt{\string\xunderh}}
  [rr]!{*\txt\small\tt{\string\xunoverh}}
 "O" !{+(0,-9)}
       !{\vover<>|>>><{x}|{y}>{z}}    
  [rur]!{\vunder<>|>>><{x}|{y}>{z}}
  [rur]!{\vunover<>|>>><{x}|{y}>{z}} 
 [urrr]!{\hover<>|>>><{x}|{y}>{z}}
    [r]!{\hunder<>|>>><{x}|{y}>{z}}     
    [r]!{\hunover<>|>>><{x}|{y}>{z}}
 "O" !{+(.5,-10.5)}
      !{*\txt\small\tt{\string\vover}} 
  [rr]!{*\txt\small\tt{\string\vunder}} 
  [rr]!{*\txt\small\tt{\string\vunover}}
 [rrr]!{*\txt\small\tt{\string\hover}} 
  [rr]!{*\txt\small\tt{\string\hunder}}
  [rr]!{*\txt\small\tt{\string\hunover}}
}$$
\caption{knot crossings with orientations and label positions}%
??=[f.crossings]
\end{figure*}

\noindent
Figure~??[f.crossings] displays the orientation on all the crossings, 
grouping them into subfamilies consisting of right-handed, left-handed
and non-crossings. Also indicated are the default positions for labels 
and arrow-tips; each piece uses the same code for tips and labels, 
e.g. |\vover<>||>>><{x}||{y}>{z}|.

The |\x...| crossings do not stack easily since their tangents are at 
$45^\circ$ to the coordinate axes. It is the last letter in the name
which denotes whether the particular crossing is vertical or horizontal.
On the other hand |\vover|\,, |\vunder| etc. stack vertically on top of 
a |\vcross|\,, |\vtwist| etc.; similarly |\hover| stacks at the left 
of |\hcross|\,, |\htwist| etc. 

\begin{code}
$$\xy 0;/r1pc/:
 ,{\vunder\vtwist\vtwist\vunder-}\endxy
\qquad\qquad\qquad \xy 0;/r1pc/:+(0,-1.5)
 ,{\hover\hcross\hcross\hover-}\endxy$$
\end{code}
\docode
\displaycode

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection*{Parsing}

\DOCMODE(
\xydef@\xykparsecross@{%
 \def\xykSCALE@@{}\edef\xyknotPLACE{\xykmidPLACE@}%
 \let\xykparser@=\xykparsecross@@
 \def\xykdefaultbreak@{\let\xykbreak@=\xykforetemp@
  \def\xyknotPLACE{(.5)}\appendtoholder@}%
 \xyFN@\xykparsescale@ }

{\xyuncatcodes \catcode`@=11
\gdef\next{%
 \ifx\space@\next \expandafter\DN@\space{\xyFN@\xykparsecross@@}%
 \else\addLT@\ifx\next \addLT@\DN@{\xyprecross@}%
 \else\addGT@\ifx\next \addGT@\DN@{\xypostcross@}%
 \else\addEQ@\ifx\next \addEQ@\DN@{\xyFN@\xykparsetips@@}%
 \else\ifx|\next\DN@|{\xymidcross@}%
 \else\addAT@\ifx\next \addAT@\DN@(##1){%
  \xyadjustBREAK@i##1@@\xyFN@\xykparsecross@@}%
 \else\DN@{\afterknot@}%
 \fi\fi\fi\fi\fi\fi \next@ }}
\xylet@\xykparsecross@@=\next
\DOCMODE)

\DOCMODE(
\xydef@\xyprecross@{%
 \def\xykholder@{\xykprebreak@@}%
 \edef\xyknotPLACE{\xykprePLACE@}%
 \xyFN@\xykparsebreak@}

\xydef@\xypostcross@{%
 \def\xykholder@{\xykpostbreak@@}%
 \edef\xyknotPLACE{\xykpostPLACE@}%
 \xyFN@\xykparsebreak@}

\xydef@\xymidcross@{%
 \def\xykholder@{\xykmidbreak@@}%
 \edef\xyknotPLACE{\xykmidPLACE@}%
 \xyFN@\xykparsebreak@}
\DOCMODE)

These macros are common to most crossings.

\DOCMODE(
\xydef@\xykcross@#1{\xykcheckTIPS@
 \expandafter\toks@\expandafter{\expandafter#1\knotSTYLE}}

\xydef@\xykoverstring@{\addtotoks@{{}}\xykmidbreak@ \xyknottips@}
\xydef@\xykunderstring@{\xykprebreak@ \xykpostbreak@ \xyknottips@}
\DOCMODE)

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection*{``cross'' crossings:}

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The initialisation\dots

\DOCMODE(
\xydef@\vcross{\begingroup\def\afterknot@{\xyvcross}%
 \def\xykprePLACE@{(.1)}\def\xykpostPLACE@{(.9)}%
 \def\xykmidPLACE@{(.9)}\def\xykpostbreak@@{|\knothole}%
 \xyFN@\xykparsecross@ }%
\xydef@\hcross{\begingroup\def\afterknot@{\xyhcross}%
  \def\xykprePLACE@{(.1)}\def\xykpostPLACE@{(.9)}%
 \def\xykmidPLACE@{(.9)}\def\xykpostbreak@@{|\knothole}%
 \xyFN@\xykparsecross@ }%
\xydef@\vcrossneg{\begingroup\def\afterknot@{\xyvcrossneg}%
  \def\xykprePLACE@{(.1)}\def\xykpostPLACE@{(.9)}%
 \def\xykmidPLACE@{(.9)}\def\xykpostbreak@@{|\knothole}%
 \xyFN@\xykparsecross@ }%
\xydef@\hcrossneg{\begingroup\def\afterknot@{\xyhcrossneg}%
  \def\xykprePLACE@{(.1)}\def\xykpostPLACE@{(.9)}%
 \def\xykmidPLACE@{(.9)}\def\xykpostbreak@@{|\knothole}%
 \xyFN@\xykparsecross@ }%
\xydef@\vuncross{\begingroup\def\afterknot@{\xyvuncross}%
  \def\xykprePLACE@{(.1)}\def\xykpostPLACE@{(.9)}%
 \def\xykmidPLACE@{(.9)}\def\xykpostbreak@@{}%
 \xyFN@\xykparsecross@ }%
\xydef@\huncross{\begingroup\def\afterknot@{\xyhuncross}%
  \def\xykprePLACE@{(.1)}\def\xykpostPLACE@{(.9)}%
 \def\xykmidPLACE@{(.9)}\def\xykpostbreak@@{}%
 \xyFN@\xykparsecross@ }%
\DOCMODE)

The interface\dots

\DOCMODE(
\xydef@\xyvcross{\xykcross@\xykcrossv
 \xykoverstring@ \xykunderstring@ \xykhvobject@{+(0,-1)}}

\xydef@\xyhcross{\xykcross@\xykcrossh
 \xykoverstring@ \xykunderstring@ \xykhvobject@{+(1,0)}}

\xydef@\xyvcrossneg{\xykcross@\xykcrossv
 \xykunderstring@ \xykoverstring@ \xykhvobject@{+(0,-1)}}

\xydef@\xyhcrossneg{\xykcross@\xykcrossh
 \xykunderstring@ \xykoverstring@ \xykhvobject@{+(1,0)}}

\xydef@\xyvuncross{\xykcross@\xykuncrossv
 \xykoverstring@ \xykunderstring@ \xykhvobject@{+(0,-1)}}

\xydef@\xyhuncross{\xykcross@\xykuncrossh
 \xykoverstring@ \xykunderstring@ \xykhvobject@{+(1,0)}}
\DOCMODE)

The drawing code\dots

\DOCMODE(
{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\save,
 "^<",{\xykz_\ar #1@'{"_<"**{}?(.25)@+,?+(.375,0)@+,
  "_>";"^>"**{}?-(.375,0)@+,?(.25)@+}"_>"#3#4#5},
 "_<",{\xykz^\ar #2@'{"^<"**{}?(.25)@+,?+(.375,0)@+,
  "^>";"_>"**{}?-(.375,0)@+,?(.25)@+}"^>"#6#7#8}
 \restore\POS #9}}
\xylet@\xykcrossv=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\save,
 "_<",{\xykz_\ar #1@'{"_>"**{}?(.25)@+,?+(0,.375)@+,
  "^>";"^<"**{}?-(0,.375)@+,?(.25)@+}"^>"#3#4#5},
 "_>",{\xykz^\ar #2@'{"_<"**{}?(.25)@+,?+(0,.375)@+,
  "^<";"^>"**{}?-(0,.375)@+,?(.25)@+}"^<"#6#7#8}
 \restore\POS #9}}
\xylet@\xykcrossh=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\save,
 "^<",{\xykz^\ar #1@'{"_<"**{}?(.25)@+,?(.375)+(.375,0)@+,
  "^>";"_>"**{}?(.375)-(.375,0)@+,?(.25)@+}"^>"#3#4#5},
 "_<",{\xykz^\ar #2@'{"^<"**{}?(.25)@+,?(.375)+(.375,0)@+,
  "_>";"^>"**{}?(.375)-(.375,0)@+,?(.25)@+}"_>"#6#7#8}
 \restore\POS #9}}
\xylet@\xykuncrossv=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\save,
 "_<",{\xykz^\ar #1@'{"_>"**{}?(.25)@+,?(.375)+(0,.375)@+,
  "^<";"^>"**{}?(.375)-(0,.375)@+,?(.25)@+}"^<"#3#4#5},
 "_>",{\xykz^\ar #2@'{"_<"**{}?(.25)@+,?(.375)+(0,.375)@+,
  "^>";"^<"**{}?(.375)-(0,.375)@+,?(.25)@+}"^>"#6#7#8}
 \restore\POS #9}}
\xylet@\xykuncrossh=\next
\DOCMODE)

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection*{``over'' crossings:}

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The initialisation\dots

\DOCMODE(
\xydef@\vover{\begingroup\def\afterknot@{\xyvover}%
 \def\xykprePLACE@{(.2)}\def\xykpostPLACE@{(.9)}%
 \def\xykmidPLACE@{(.8)}\def\xykpostbreak@@{|(.725)\knothole}%
 \xyFN@\xykparsecross@ }%

\xydef@\hover{\begingroup\def\afterknot@{\xyhover}%
 \def\xykprePLACE@{(.2)}\def\xykpostPLACE@{(.9)}%
 \def\xykmidPLACE@{(.8)}\def\xykpostbreak@@{|(.725)\knothole}%
 \xyFN@\xykparsecross@ }%

\xydef@\vunder{\begingroup\def\afterknot@{\xyvunder}%
 \def\xykprePLACE@{(.1)}\def\xykpostPLACE@{(.8)}%
 \def\xykmidPLACE@{(.2)}\def\xykpostbreak@@{|(.275)\knothole}%
 \xyFN@\xykparsecross@ }%

\xydef@\hunder{\begingroup\def\afterknot@{\xyhunder}%
 \def\xykprePLACE@{(.1)}\def\xykpostPLACE@{(.8)}%
 \def\xykmidPLACE@{(.2)}\def\xykpostbreak@@{|(.275)\knothole}%
 \xyFN@\xykparsecross@ }%

\xydef@\vunover{\begingroup\def\afterknot@{\xyvunover}%
 \def\xykprePLACE@{(.1)}\def\xykpostPLACE@{(.9)}%
 \def\xykmidPLACE@{(.8)}\def\xykpostbreak@@{}%
 \xyFN@\xykparsecross@ }%

\xydef@\hunover{\begingroup\def\afterknot@{\xyhunover}%
 \def\xykprePLACE@{(.1)}\def\xykpostPLACE@{(.9)}%
 \def\xykmidPLACE@{(.8)}\def\xykpostbreak@@{}%
 \xyFN@\xykparsecross@ }%


\xydef@\xoverv{\begingroup\def\afterknot@{\xyxoverv}%
 \def\xykprePLACE@{(.15)}\def\xykpostPLACE@{(.85)}%
 \def\xykmidPLACE@{(.15)}\def\xykpostbreak@@{|\knothole}%
 \xyFN@\xykparsecross@ }%

\xydef@\xunderv{\begingroup\def\afterknot@{\xyxunderv}%
 \def\xykprePLACE@{(.15)}\def\xykpostPLACE@{(.85)}%
 \def\xykmidPLACE@{(.15)}\def\xykpostbreak@@{|\knothole}%
 \xyFN@\xykparsecross@ }%

\xydef@\xoverh{\begingroup\def\afterknot@{\xyxoverh}%
 \def\xykprePLACE@{(.15)}\def\xykpostPLACE@{(.85)}%
 \def\xykmidPLACE@{(.15)}\def\xykpostbreak@@{|\knothole}%
 \xyFN@\xykparsecross@ }%

\xydef@\xunderh{\begingroup\def\afterknot@{\xyxunderh}%
 \def\xykprePLACE@{(.15)}\def\xykpostPLACE@{(.85)}%
 \def\xykmidPLACE@{(.85)}\def\xykpostbreak@@{|\knothole}%
 \xyFN@\xykparsecross@ }%

\xydef@\xunover{\begingroup\def\afterknot@{\xyxunover}%
 \def\xykprePLACE@{(.15)}\def\xykpostPLACE@{(.85)}%
 \def\xykmidPLACE@{(.15)}\def\xykpostbreak@@{}%
 \xyFN@\xykparsecross@ }%

\xydef@\xunoverv{\begingroup\def\afterknot@{\xyxunoverv}%
 \def\xykprePLACE@{(.2)}\def\xykpostPLACE@{(.8)}%
 \def\xykmidPLACE@{(.2)}\def\xykpostbreak@@{}%
 \xyFN@\xykparsecross@ }%

\xydef@\xunoverh{\begingroup\def\afterknot@{\xyxunoverh}%
 \def\xykprePLACE@{(.2)}\def\xykpostPLACE@{(.8)}%
 \def\xykmidPLACE@{(.8)}\def\xykpostbreak@@{}%
 \xyFN@\xykparsecross@ }%
\DOCMODE)

The interface\dots

\DOCMODE(
\xydef@\xyvover{\xykcross@\xykoverv
 \xykoverstring@ \xykunderstring@ \xykhvobject@{+(0,-1)}}

\xydef@\xyhover{\xykcross@\xykoverh
 \xykoverstring@ \xykunderstring@ \xykhvobject@{+(1,0)}}

\xydef@\xyvunder{\xykcross@\xykoverv
 \xykunderstring@ \xykoverstring@ \xykhvobject@{+(0,-1)}}

\xydef@\xyhunder{\xykcross@\xykoverh
 \xykunderstring@ \xykoverstring@ \xykhvobject@{+(1,0)}}

\xydef@\xyvunover{\xykcross@\xykunoverv
 \xykoverstring@ \xykunderstring@ \xykhvobject@{+(0,-1)}}

\xydef@\xyhunover{\xykcross@\xykunoverh
 \xykoverstring@ \xykunderstring@ \xykhvobject@{+(1,0)}}


\xydef@\xyxoverv{\xykcross@\xykxoverv
 \xykoverstring@ \xykunderstring@ \xykhvobject@{+(0,-1)}}

\xydef@\xyxunderv{\xykcross@\xykxoverv
 \xykunderstring@ \xykoverstring@ \xykhvobject@{+(0,-1)}}

\xydef@\xyxoverh{\xykcross@\xykxoverh
 \xykoverstring@ \xykunderstring@ \xykhvobject@{+(1,0)}}

\xydef@\xyxunderh{\xykcross@\xykxoverh
 \xykunderstring@ \xykoverstring@ \xykhvobject@{+(1,0)}}

\xydef@\xyxunover{\xykcross@\xykxoverv
 \xykoverstring@ \xykunderstring@ \xykhvobject@{+(1,0)}}

\xydef@\xyxunoverv{\xykcross@\xykxunoverv
 \xykoverstring@ \xykunderstring@ \xykhvobject@{+(0,-1)}}

\xydef@\xyxunoverh{\xykcross@\xykxunoverh
 \xykoverstring@ \xykunderstring@ \xykhvobject@{+(1,0)}}

\DOCMODE)

The drawing code\dots

\DOCMODE(
{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\save,
 "^<",{\xykz^\ar #1@'{"^>"**{}?(.25)@+,?(.625)+(0,-.125)@+,
  "_>";"^>"**{}?(.625)-(.125,0)@+,?(.25)@+}"_>"#3#4#5},
 "_<",{\xykz^\ar #2@'{"^<"**{}?(.25)@+,?(.625)+(.125,0)@+,
  "^>";"^<"**{}?(.625)-(0,.125)@+,?(.25)@+}"^>"#6#7#8},\restore\POS#9}}
\xylet@\xykoverv=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\save,
 "_<",{\xykz^\ar #1@'{"^<"**{}?(.25)@+,?(.625)+(.125,0)@+,
  "^>";"^<"**{}?(.625)-(0,.125)@+,?(.25)@+}"^>"#3#4#5},
 "_>",{\xykz^\ar #2@'{"_<"**{}?(.25)@+,?(.625)+(0,.125)@+,
  "^<";"_<"**{}?(.625)+(.125,0)@+,?(.25)@+}"^<"#6#7#8},\restore\POS#9}}
\xylet@\xykoverh=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\save,
 "^<",{\xykz^\ar #1@'{"^>"**{}?(.25)@+,?-(0,.175)@+,?(.75)@+}
  "^>"#3#4#5},
 "_<",{\xykz^\ar #2@'{"^<"**{}?(.25)@+,?+(.125,0)@+,
   "_>"**{}?+(0,.7)@+,"_>";"^>"**{}?-(.125,0)@+,?(.25)@+}
  "_>"#6#7#8},\restore\POS#9}}
\xylet@\xykunoverv=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\save,
 "_<",{\xykz^\ar #1@'{"^<"**{}?(.25)@+,?+(.175,0)@+,?(.75)@+}
  "^<"#3#4#5},
 "_>",{\xykz^\ar #2@'{"_<"**{}?(.25)@+,?+(0,.125)@+,
   "^>"**{}?-(.7,0)@+,"^>";"^<"**{}?-(0,.125)@+,?(.25)@+}
  "^>"#6#7#8},\restore\POS#9}}
\xylet@\xykunoverh=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\save,
 "^<";"^>"**{}?="^_","_<";"_>"**{}?;"^_",**{}?="^_",
 "^>",{\xykz^\ar #1@'{"^_"@+}"_<"#3#4#5},
 "^<",{\xykz_\ar #2@'{"^_"@+}"_>"#6#7#8},\restore\POS#9}}
\xylet@\xykxoverv=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\save,
 "^<";"_<"**{}?="^_","^>";"_>"**{}?;"^_",**{}?="^_",
 "^<",{\xykz_\ar #1@'{"^_"@+}"_>"#3#4#5},
 "_<",{\xykz_\ar #2@'{"^_"@+}"^>"#6#7#8},\restore\POS#9}}
\xylet@\xykxoverh=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\save,
 "^<";"^>"**{}?="^_","_<";"_>"**{}?;"^_",**{}?="^_",
 "^>",{\xykz^\ar #1@'{"^_"@+}"_>"#3#4#5},
 "^<",{\xykz^\ar #2@'{"^_"@+}"_<"#6#7#8},\restore\POS#9}}
\xylet@\xykxunoverv=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\save,
 "^<";"_<"**{}?="^_","^>";"_>"**{}?;"^_",**{}?="^_",
 "^<",{\xykz^\ar #1@'{"^_"@+}"^>"#3#4#5},
 "_<",{\xykz^\ar #2@'{"^_"@+}"_>"#6#7#8},
  \restore\POS#9}}
\xylet@\xykxunoverh=\next
\DOCMODE)

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection*{``twist'' crossings:}

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The initialisation\dots

\DOCMODE(
\xydef@\vtwist{\begingroup\def\afterknot@{\xyvtwist}%
 \def\xykprePLACE@{(.1)}\def\xykpostPLACE@{(.9)}%
 \def\xykmidPLACE@{(.9)}\def\xykpostbreak@@{|\knothole}%
 \xyFN@\xykparsecross@ }%
\xydef@\htwist{\begingroup\def\afterknot@{\xyhtwist}%
 \def\xykprePLACE@{(.1)}\def\xykpostPLACE@{(.9)}%
 \def\xykmidPLACE@{(.9)}\def\xykpostbreak@@{|\knothole}%
 \xyFN@\xykparsecross@ }%
\xydef@\vtwistneg{\begingroup\def\afterknot@{\xyvtwistneg}%
 \def\xykprePLACE@{(.1)}\def\xykpostPLACE@{(.9)}%
 \def\xykmidPLACE@{(.9)}\def\xykpostbreak@@{|\knothole}%
 \xyFN@\xykparsecross@ }%
\xydef@\htwistneg{\begingroup\def\afterknot@{\xyhtwistneg}%
 \def\xykprePLACE@{(.1)}\def\xykpostPLACE@{(.9)}%
 \def\xykmidPLACE@{(.9)}\def\xykpostbreak@@{|\knothole}%
 \xyFN@\xykparsecross@ }%
\xydef@\vuntwist{\begingroup\def\afterknot@{\xyvuntwist}%
 \def\xykprePLACE@{(.15)}\def\xykpostPLACE@{(.85)}%
 \def\xykmidPLACE@{(.85)}\def\xykpostbreak@@{}%
 \xyFN@\xykparsecross@ }%
\xydef@\huntwist{\begingroup\def\afterknot@{\xyhuntwist}%
 \def\xykprePLACE@{(.15)}\def\xykpostPLACE@{(.85)}%
 \def\xykmidPLACE@{(.85)}\def\xykpostbreak@@{}%
 \xyFN@\xykparsecross@ }%
\DOCMODE)

The interface\dots

\DOCMODE(
\xydef@\xyvtwist{\xykcross@\xyktwistv
 \xykoverstring@ \xykunderstring@ \xykhvobject@{+(0,-1)}}

\xydef@\xyvtwistneg{\xykcross@\xyktwistv
 \xykunderstring@ \xykoverstring@ \xykhvobject@{+(0,-1)}}

\xydef@\xyvuntwist{\xykcross@\xykuntwistv
 \xykoverstring@ \xykunderstring@ \xykhvobject@{+(0,-1)}}

\xydef@\xyhtwist{\xykcross@\xyktwisth
 \xykoverstring@ \xykunderstring@ \xykhvobject@{+(1,0)}}

\xydef@\xyhuntwist{\xykcross@\xykuntwisth
 \xykoverstring@ \xykunderstring@ \xykhvobject@{+(1,0)}}

\xydef@\xyhtwistneg{\xykcross@\xyktwisth
 \xykunderstring@ \xykoverstring@ \xykhvobject@{+(1,0)}}
\DOCMODE)

The drawing code\dots

\DOCMODE(
{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\save,
 "^>",{\xykz_\ar #1@'{"_>"**{}?(.25)@+,?-(.375,0)@+,
  "_<";"^<"**{}?+(.375,0)@+,?(.25)@+}"_<"#3#4#5},
 "^<",{\xykz^\ar #2@'{"_<"**{}?(.25)@+,?+(.375,0)@+,
  "_>";"^>"**{}?+(-.375,0)@+,?(.25)@+}"_>"#6#7#8}
 \restore\POS #9}}
\xylet@\xyktwistv=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\save,
 "^>",{\xykz^\ar #1@'{"_>"**{}?(.125)@+,?(.375)-(.25,0)@+,
  ?(.625)-(.25,0)@+,?(.875)@+}"_>"#3#4#5},
 "^<",{\xykz^\ar #2@'{"_<"**{}?(.125)@+,?(.375)+(.25,0)@+,
  ?(.625)+(.25,0)@+,?(.875)@+}"_<"#6#7#8}
 \restore\POS #9}}
\xylet@\xykuntwistv=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\save,
 "^<",{\xykz_\ar #1@'{"^>"**{}?(.25)@+,?+(0,-.375)@+,
  "_>";"_<"**{}?+(0,.375)@+,?(.25)@+}"_>"#3#4#5},
 "_<",{\xykz^\ar #2@'{"_>"**{}?(.25)@+,?+(0,.375)@+,
  "^>";"^<"**{}?-(0,.375)@+,?(.25)@+}"^>"#6#7#8}
 \restore\POS #9}}
\xylet@\xyktwisth=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\save,
 "^<",{\xykz^\ar #1@'{"^>",**{}?(.125)@+,?(.375)+(0,-.25)@+,
  ?(.625)+(0,-.25)@+,?(.875)@+}"^>"#3#4#5},
 "_<",{\xykz^\ar #2@'{"_>",**{}?(.125)@+,?(.375)+(0,.25)@+,
  ?(.625)+(0,.25)@+,?(.875)@+}"_>"#6#7#8},\restore\POS #9}}
\xylet@\xykuntwisth=\next

\DOCMODE)

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\note??=[scale the knot piece]
The above examples also show how to use |-| to get the mirror-image of a 
particular crossing. Any numerical scale factor can be used by enclosing 
it within |[..]| e.g. |[2.3]| scaling a single piece without affecting 
the rest of the picture. The scale-factor {\em must\/} occur before any
label or arrow-tip specifiers, see below). Vertical crossings remain 
vertical under scalings; the current <pos> still moves by 1 coordinate 
unit in the `down' direction. Similarly horizontal crossings remain 
horizontal. The single character |-| is a shorthand version for |[-1]|,
effectively giving a half-turn rotation in a rectangular basis.

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\DOCMODE(
\xydef@\xykparsescale@{%
 \ifx\space@\next \expandafter\DN@\space{\xyFN@\xykparsescale@}%
 \else\addDASH@\ifx\next \addDASH@\DN@{\def\xykSCALE@@{-1}%
  \xyFN@\xykparser@}%
 \else\ifx[\next \DN@[##1]{\dimen@=##1\p@
  \ifdim\dimen@=\z@\xywarning@{\xykscaleerror@}%
  \else\edef\xykSCALE@@{\expandafter\removePT@\the\dimen@}\fi
  \xyFN@\xykparser@}%
 \else\ifx~\next \DN@~##1##2##3##4{%
  \def\xykSCALE@@{@{##1}{##2}{##3}{##4}}\xyFN@\xykparser@}%
 \else\DN@{\xyFN@\xykparser@}%
 \fi\fi\fi\fi \next@}

\xydef@\xykscaleerror@{Ignoring zero scale factor}
\DOCMODE)

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\note??=[alter size and shape]
A knot-piece need not be rectangular. By specifying
|~|<pos>$_1$<pos$_2$><pos$_3$><pos$_4$> the four corners |UL|, |UR|, 
|DL|, |DR| are set to the given <pos>s respectively. 
The local basis is established so that 
\begin{eqnarray*}
r\hbox{--hop}&\!\!\leftrightarrow\!\!&
{\textstyle{1\over2}}\bigl(\hbox{<pos$_2$>}-\hbox{<pos$_1$>}
+\hbox{<pos$_4$>} -\hbox{<pos$_3$>}\bigr) \\
u\hbox{--hop}&\!\!\leftrightarrow\!\!&
{\textstyle{1\over2}}\bigl(\hbox{<pos$_1$>}-\hbox{<pos$_3$>}
+\hbox{<pos$_2$>} -\hbox{<pos$_1$>}\bigr)\;.
\end{eqnarray*}

\note??=[adjust hole]
With a non-rectangularly shaped piece it will usually be necessary to 
adjust the place where the `hole' occurs in the `under' string. This is
done by specifying |@(|<num>|)|, with $0\le <num>\le1$ being the parameter
value of the new location for the hole.

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\DOCMODE(
%\xydef@\xykobject@{%
% \ifx\xykSCALE@@\empty\DN@{\xykkeepscale@}%
% \else \DN@{\xyksavescale@{}}%
% \fi \next@ }

\xydef@\xykhobject@{\aftergroup\xykhadjust@
 \ifx\xykSCALE@@\empty\DN@{\xykkeepscale@}%
 \else \DN@{\xyksavescale@{}}%
 \fi \next@ }

\xydef@\xykvobject@{\aftergroup\xykvadjust@
 \ifx\xykSCALE@@\empty\DN@{\xykkeepscale@}%
 \else \DN@{\xyksavescale@{}}%
 \fi \next@ }

\xydef@\xykhvobject@{\aftergroup\xykhvadjust@
 \ifx\xykSCALE@@\empty\DN@{\xykkeepscale@}%
 \else \DN@{\xyksavescale@{}}%
 \fi \next@ }

\xydef@\xykhuobject@{\aftergroup\xykhuadjust@
 \ifx\xykSCALE@@\empty\DN@{\xykkeepscale@}%
 \else \DN@{\xyksavescale@{}}%
 \fi \next@ }
\DOCMODE)

\DOCMODE(
\xydef@\xyksavescale@#1#2{%
 \expandafter\ifx\the\Edge@c\zeroEdge\relax \def\xyksaveEdgec{}%
 \else
  \edef\xyksaveEdgec{\noexpand\Edge@c={\expandafter\noexpand\the\Edge@c}%
   \L@c=\the\L@c \R@c=\the\R@c \D@c=\the\D@c \U@c=\the\U@c}%
 \fi
%
 \edef\next@##1##2{\endgroup
  \def\noexpand\xyksaveEdgec{}%
  \def\noexpand\xykSCALE@@\noexpand{##1}%
  \def\noexpand\xykjoincontrol\noexpand{\xykjoincontrol}%
  \noexpand\toks@\noexpand{##2}%
  \noexpand\xy@@{\def\noexpand\xyksaveEdgec{}%
   \def\noexpand\xykSCALE@@{##1}%
   \def\noexpand\xykjoincontrol{\xykjoincontrol}}}%
 \expandafter\expandafter\expandafter\next@
 \expandafter\expandafter\expandafter{\expandafter\xykSCALE@@\expandafter}%
 \expandafter{\the\toks@}%
%
 \xykrescale@ #1\the\toks@\restore\POS#2}

\xydef@\xykkeepscale@{%
 \edef\next@##1{\endgroup \noexpand\toks@={##1}%
  \def\noexpand\xykjoincontrol{\xykjoincontrol}}%
 \expandafter\next@\expandafter{\the\toks@}%
 \def\xykSCALE@@{}\def\xyksaveEdgec{}%
 \xy@@{\def\xykSCALE@@{}\def\xyksaveEdgec{}}%
 \edef\next@{\def\noexpand\xykjoincontrol{\xykjoincontrol}}%
 \expandafter\xy@@\expandafter{\next@}%
 \xyknoflexscale@ \the\toks@}
\DOCMODE)

\DOCMODE(
\xydef@\xykrescale@{\expandafter\xykcheckscale@\xykSCALE@@ !}%

\xydef@\xykcheckscale@#1#2!{%
 \expandafter\DN@\expandafter{\codeof{#1}}%
 \expandafter\DNii@\expandafter{\codeof{@}}%
 \ifx\next@\nextii@\relax\DN@{\xykflexiscale@#2%
 \xy@@{\divide\X@xbase2\divide\X@ybase2\divide\Y@xbase2\divide\Y@ybase2%
  \def\xykSCALE@@{}}\def\xykSCALE@@{}%
  \expandafter\toks@\expandafter{\afterFLEX@}}%
 \else\DN@{\save \xykrescale@@}\fi \next@ }

\xydef@\xykflexiscale@{%
 \expandafter\def\expandafter\afterFLEX@\expandafter{\the\toks@}%
 \xykflexiscale@@ }

\xydef@\xykrescale@@{%
 \edef\next@{\X@xbase=\xykSCALE@@\X@xbase \X@ybase=\xykSCALE@@\X@ybase
  \Y@xbase=\xykSCALE@@\Y@xbase \Y@ybase=\xykSCALE@@\Y@ybase}%
 \expandafter\xy@@\expandafter{\next@}%
 \ifdim\xykSCALE@@\p@<\z@\relax \DN@{\xyknegflexscale@}%
 \else\DN@{\xyknoflexscale@}\fi \next@ }
\DOCMODE)


\DOCMODE(
{\xyuncatcodes \gdef\next#1#2#3#4{\save="@",
  #1,="^<",\POS"@",#2,="^>",\POS"@",#3,="_<",\POS"@",#4,="_>",
 (0,0);"^>"-"^<"+"_>"-"_<":"^<"-"_<"+"^>"-"_>"::}}
\xylet@\xykflexiscale@@=\next

{\xyuncatcodes \gdef\next{\save ="^<",
 +(1,0)="^>",-(1,1)="_<",+(1,0)="_>"\restore }}
\xylet@\xyknoflexscale@@=\next
\xylet@\xyknoflexscale@=\next

{\xyuncatcodes \gdef\next{\save ="_>",
 +(-1,1)="^<",+(1,0)="^>",-(1,1)="_<"\restore }}
\xylet@\xyknegflexscale@@=\next
\xylet@\xyknegflexscale@=\next

{\xyuncatcodes \gdef\next{\save ="_<",
 +(1,0)="_>",+(-1,1)="^<",+(1,0)="^>"\restore }}
\xylet@\xykhscale@@=\next

{\xyuncatcodes \gdef\next{\save ="^>",
 -(1,0)="^<",+(1,-1)="_>",-(1,0)="_<"\restore }}
\xylet@\xykvscale@@=\next
\DOCMODE)


\DOCMODE(
\xydef@\xykhadjust@{%
 \let\xyknoflexscale@=\xykhscale@@
 \let\xyknegflexscale@=\xyknegflexscale@@ }

\xydef@\xykvadjust@{%
 \let\xyknoflexscale@=\xyknoflexscale@@
 \let\xyknegflexscale@=\xykhscale@@ }

\xydef@\xykhvadjust@{%
 \let\xyknoflexscale@=\xyknoflexscale@@
 \let\xyknegflexscale@=\xyknegflexscale@@ }

\xydef@\xykhuadjust@{%
 \let\xyknoflexscale@=\xykhscale@@
 \let\xyknegflexscale@=\xykvscale@@ }
\DOCMODE)

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\note??=[arrowtips, breaks and labels]
The |knot| feature allows for the easy placement of the following objects 
along the strings of a crossing:\par
\begin{itemize}
\item labels on the strings;
\item arrowheads for direction or orientation;
\item holes in strings, allowing another string to be drawn passing over.
\end{itemize}

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\note ??=[which string]
The characters |<|, |>| and |||| are used to indicate to which string 
portion the object is associated; with |||| denoting the string which
crosses the other, while |<| and |>| denote the initial and final 
portions of the `crossed' string. 

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\DOCMODE(
\xydef@\xykprebreak@{%
 \ifx\xykprebreak@@\empty\addtotoks@{{}}\else
  \expandafter\addtotoks@\expandafter{\expandafter{\xykprebreak@@}}\fi}

\xydef@\xykpostbreak@{%
 \ifx\xykpostbreak@@\empty\addtotoks@{{}}\else
  \expandafter\addtotoks@\expandafter{\expandafter{\xykpostbreak@@}}\fi}

\xydef@\xykmidbreak@{%
 \ifx\xykmidbreak@@\empty\addtotoks@{{}}\else
  \expandafter\addtotoks@\expandafter{\expandafter{\xykmidbreak@@}}\fi}
\DOCMODE)

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\note ??=[simple label]
A simple label enclosed in braces, for example |\vcross>{x}|, is set in 
math-mode using the |\labelstyle|, at a pre-determined place on the string 
portion, shifted in either the `above' or `below' direction from the curve 
at this point. (For each crossing depicted in figure~??[f.crossings] only 
default values are used for the place and shift-direction.)

\note ??=[knot object]
If the first character within the braces |{..}| is |*| e.g.
|\htwist>{*|<object>|}|, then a general <object> may be placed as a label. 
Furthermore if the first character is |^| or |_| or ||||, then
the interpretation is, e.g. |\vtwist<{^|<anchor><it>|}|, 
as in ??g[xyarrow.doc:f.arrow] 
to place <it> as a label along an |\ar| of the |arrow| feature. 

\note ??=[orientation]
A second character |<| or |>| specifies that an arrowhead should appear 
at the pre-determined place on the chosen string.
Here |>| denotes an arrowhead pointing with the natural orientation,
while |<| points against. Due to the curvature of the strings, it is
usually best to |\UseComputerModernTips| rather than normal arrow-tips.

\note ??=[hole in the string]
To generate a `hole' use |\knothole|, or simply |\khole|, as  following 
token. This generates a `break', in the sense of 
??g[xyarrow.doc:break with <it>]. 
Indeed such a `hole' is used to separate the two portions of the `crossed' 
string. Default size for the hole is 5pt, which is alterable via 
|\knotholesize{|<dimen>|}|; normally used to set the size for {\em all}\/ 
holes in a diagram. 

\note ??=[null-break]
If the resulting |\khole| is either too large or perhaps non-existent,
this could be due to a technicality in the way breaks in curves are 
handled. This problem should not occur with the standard crossings, using
a rectangular basis, but it may occur with non-rectangular bases.
An easy `fix' is to include an extra {\em null-break}\ on the string, 
using |<|||, |>||| or ||||||, which should place the zero-sized break at
parameter value .5 on the curve. The specification should precede a 
|\khole| at a higher parameter value, or come after one at a lower value.

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The aim of the above `fix' is to position the null-break as close as 
possible to where the curve is farthest from the line joining its 
end-points, which is usually at parameter value |(.5)|.

\DOCMODE(
\xydef@\xykparsebreak@{%
 \ifx\space@\next \expandafter\DN@\space{\xyFN@\xykparsebreak@}%
 \else \addLT@\ifx\next 
  \expandafter\appendtoholder@\xykbacktemp@\next@
  \addLT@\DN@{\xyFN@\xykparser@}%
 \else \addGT@\ifx\next 
  \expandafter\appendtoholder@\xykforetemp@\next@
  \addGT@\DN@{\xyFN@\xykparser@}%
 \else \ifx|\next
  \expandafter\appendtoholer@\xyknulltemp@\next@
  \DN@|{\xyFN@\xykparser@}%
 \else \ifx\next \bgroup\DN@##1{\checkgroupbreak@##1@}%
 \else \ifx (\next\DN@(##1){\xyadjustknotPLACE@i##1@@%
  \xyFN@\xykparsebreak@}%
 \else \ifx\next \knothole@\DN@##1{\expandafter\appendtoholder@@
   \expandafter{\expandafter|\xyknotPLACE\knothole}\next@
  \xyFN@\xykparser@}%
 \else 
  \expandafter\xykdefaultbreak@\xykbreak@\next@
  \DN@{\xyFN@\xykparser@}%
 \fi\fi\fi\fi\fi\fi\fi \next@ }
\DOCMODE)

Templates for arrowheads.

\DOCMODE(
{\xyuncatcodes \gdef\next{^>*\dir{>}}}
\xylet@\xykforetips@=\next

{\xyuncatcodes \gdef\next{^<*\dir{<}}}
\xylet@\xykbacktips@=\next

{\xyuncatcodes \gdef\next{*!/-1pt/=0\dir{>}}}
\xylet@\xykforetemp@=\next

{\xyuncatcodes \gdef\next{*!/1pt/=0\dir{<}}}
\xylet@\xykbacktemp@=\next

{\xyuncatcodes \gdef\next{*=<\xykholesize>[o]{}}}
\xylet@\xykholetemp@=\next
\xydef@\knothole@{\hbox to\xykholesize{%
 \dimen@=\xykholesize \divide\dimen@\tw@
 \hfill \vrule height\dimen@ depth\dimen@ width\z@ }}
\xylet@\knothole=\knothole@
\xylet@\khole=\knothole@

{\xyuncatcodes \gdef\next{*=<0pt>{}}}
\xylet@\xyknulltemp@=\next
\xylet@\xykbreak@=\xyknulltemp@
\DOCMODE)

When the next token is a begin-group character |{| then look at the first
token inside the group. If it is |^|,|_| or |||| then the group contents
will be an <anchor> for user-chosen places along the string.
Otherwise the place is taken to be the already specified |\xyknotPLACE|
and the group contains a label, to be set in the usual way for labels, 
using |_| due to the orientation of the curve. 
However if the first token is |*| then the user is assumed to be 
dropping his own label, possibly with <object> modifiers.

\DOCMODE(
\xydef@\checkgroupbreak@#1#2@{%
 \ifx|#1\DN@{\appendtoholder@@#1#2\next@ \xyFN@\xykparser@}%
 \else \ifx_#1\DN@{\appendtoholder@@_#2\next@ \xyFN@\xykparser@}%
 \else \ifx^#1\DN@{\appendtoholder@@^#2\next@ \xyFN@\xykparser@}%
 \else 
 \def\next##1\next{\expandafter\DNii@\expandafter{\xyk@@##1}}%
 \expandafter\next\xyknotPLACE\next
 \ifx*#1%
  \expandafter\DN@\expandafter{%
    \expandafter\appendtoholder@@\nextii@#1#2\next@ \xyFN@\xykparser@}%
 \else %\expandafter\DNii@\expandafter{\expandafter\xyk@@\xyknotPLACE}%
   \expandafter\DN@\expandafter{%
    \expandafter\appendtoholder@@\nextii@{#1#2}\next@ \xyFN@\xykparser@}%
 \fi\fi\fi\fi \next@ }
\DOCMODE)

\DOCMODE(
{\xyuncatcodes \gdef\next{%
 \expandafter{\expandafter^\xyknotPLACE##1}}}%
 \DN@{\xydef@\xykholder@@##1}\DNii@{\expandafter\toks@}%
 \expandafter\expandafter\expandafter\next@
 \expandafter\expandafter\expandafter{\expandafter\nextii@\next}

\xydef@\appendtoholder@#1\next@{\xykholder@@{#1}%
 \DN@##1{\expandafter\DN@\expandafter{\xykholder@##1}}%
 \expandafter\next@\expandafter{\the\toks@}\appendtoholder@@@ }

\xydef@\appendtoholder@@#1\next@{%
 \DN@##1\next@{\expandafter\DN@\expandafter{\xykholder@##1}}%
  \next@#1\next@ \appendtoholder@@@}%

\xydef@\appendtoholder@@@{%
 \expandafter\expandafter\expandafter\DN@
  \expandafter\expandafter\expandafter{\next@}%
 \DNii@{\expandafter\def\xykholder@}%
  \expandafter\nextii@\expandafter{\next@}}
\DOCMODE)


\DOCMODE(
{\xyuncatcodes \gdef\next{%
 \expandafter{\expandafter|\xyknotPLACE##1}}}%
\DN@{\xydef@\xykholer@@##1}\DNii@{\expandafter\toks@}%
 \expandafter\expandafter\expandafter\next@
 \expandafter\expandafter\expandafter{\expandafter\nextii@\next}

\xydef@\appendtoholer@#1\next@{\xykholer@@{#1}%
 \DN@##1{\expandafter\DN@\expandafter{\xykholder@##1}}%
 \expandafter\next@\expandafter{\the\toks@}\appendtoholder@@@ }
\DOCMODE)

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

??=[knot-labels]
Multiple breaks, arrow-heads and labels may be specified along the 
two strings of a crossing; simply place their specifications one after
another; e.g. |<>||>>><{x}||{y}>{z}| was used in figure~??[f.crossings].

The only proviso is that all `breaks' along a single strand must occur 
with increasing order of parameter position. On the `crossed' string 
this includes the automatic `hole' to create space for the other string. 
Hence it is advisable to use just the |(+..)| and |(-..)| variants for
small adjustments, and to keep these correctly ordered.

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\DOCMODE(
\xydef@\xykz#1{\let\xyk@@=#1\xy@@{\let\xyk@@=#1}}
{\xyuncatcodes \global\let\next=_}
\xylet@\xyk@@=\next
\DOCMODE)

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

??=[adjustment]
Adjustment of position along the strings can be achieved using a <factor>,
as in |\vover||(+.1)>|. Allowed syntax is |(|<sign><num>|)| where <sign> 
is |+| or |-| to increment or decrement from the pre-defined value. 
Also allowable are |=| or <empty> to set the parameter position to <num>, 
which must lie between 0 and 1 to have any meaning. 

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This parses adjustments of the |\xyknotPLACE|.

\DOCMODE(
\xydef@\xyadjustknotPLACE@i{%
 \ifx\xyknotPLACE\empty\DN@{\dimen@=.5\p@}%
 \else\DN@{\expandafter\xygetknotPLACE@@\xyknotPLACE}%
 \fi \next@ \relax \let\xywhichknotPLACE@@=\xysetknotPLACE@@
 \xyFN@\xyadjustknotPLACE@}

\xydef@\xyadjustknotPLACE@{\expandafter\ifx\space@\next
  \expandafter\DN@\space{\xyFN@\xyadjustknotPLACE@}%
 \else\addDASH@\ifx\next\DN@{\xyadjustknotPLACE@@}%
 \else\addPLUS@\ifx\next\DN@{\xyadjustknotPLACE@@}%
 \else\addEQ@\ifx\next\addEQ@\DN@{\dimen@=\z@ \xyadjustknotPLACE@@}%
 \else \DN@{\dimen@=\z@ \xyadjustknotPLACE@@}%
 \fi\fi\fi\fi \next@}

\xydef@\xyadjustknotPLACE@@#1@@{\advance\dimen@#1\p@
 \expandafter\xysetknotPLACE@\expandafter{\the\dimen@}}

\xydef@\xysetknotPLACE@#1{%
 \expandafter\xywhichknotPLACE@@\expandafter{\removePT@#1}}

{\xyuncatcodes\gdef\next#1{\def\xyknotPLACE{(#1)}}}%
\xylet@\xysetknotPLACE@@=\next

{\xyuncatcodes \catcode`@=11
 \gdef\next(#1){\DN@\next@{#1}\ifx\next@\empty
  \dimen@=\z@\else\dimen@=#1\p@ \fi}}%
\xylet@\xygetknotPLACE@@=\next 
\DOCMODE)

These use the same mechanism to adjust parameters stored in 
|\xykpostbreak@@| and |\xykjoincontrol@@|.

\DOCMODE(
%\xydef@\xyadjustJOIN@i#1@@{\xykjoincontrol@i{#1}}%
\xydef@\xyadjustJOIN@i{%
 \ifx\xykjoincontrol\empty\DN@{\dimen@=.75\p@}%
 \else\DN@{\expandafter\xygetknotPLACE@@\expandafter(\xykjoincontrol)}%
 \fi \next@ \relax \let\xywhichknotPLACE@@=\xysetjoincontrol@
 \xyFN@\xyadjustknotPLACE@}

%\xydef@\xyadjustBREAK@i#1@@{\def\xykpostbreak@@{|(##1)\knothole}}%
\xydef@\xyadjustBREAK@i{%
 \ifx\xykpostbreak@@\empty\DN@{\dimen@=.5\p@}%
 \else\DN@{\expandafter\xygetpostBREAK@\xykpostbreak@@}%
 \fi \next@ \relax \let\xywhichknotPLACE@@=\xysetpostbreak@
 \xyFN@\xyadjustknotPLACE@}

\xydef@\xygetpostBREAK@|#1\knothole{\DN@{#1}%
 \ifx\next@\empty\dimen@=.5\p@\else \xygetknotPLACE@@#1\fi}

\xydef@\xysetpostbreak@#1{\def\xykpostbreak@@{|(#1)\knothole}}
\xydef@\xysetjoincontrol@#1{\edef\xykjoincontrol{#1}}
\DOCMODE)

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

??=[arrowtips]
Arrowheads can also be placed at either, or both, ends of of the strings
forming a crossing. This is governed by a pair of booleans, 
initially |{FF}|.
It is changed for {\em all}\/ subsequent strings in a diagram by 
|\knottips{..}| where the recognised values are |{FF}|, |{FT}|, |{TF}| 
and |{TT}|, denoting tips (|T|) or not (|F|) at the start and end of each
string. To add arrowtips at the start of strings in a particular crossing, 
append the 2-character combination |=<|\,; similarly |=>| adds tips at the 
ends, if not already requested. The combinations |==| and |=!| specify 
both (|{TT}|) and none (|{FF}|) respectively. 
These 2-character pairs can be mixed in with any specifications for labels
and breaks, etc. Multiple pairs compound their effect; in particular
|=<=>| gives the same result as |==|, while |=!=<| is needed to change 
|{FT}| into |{TF}|.

These are best used with single pieces, as in the following equation.
%
\begin{code}
\UseComputerModernTips \knottips{FT}
\def\Conway#1{\mathord{\nabla\Bigl[\,
 \raise5pt\xybox{0;/r1pc/:#1}\,\Bigr]}}
$$
\Conway\htwist - \Conway\htwistneg
 \;=\; -z\,\Conway\huntwist $$
\end{code}
\docode
\displaycode

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\DOCMODE(
\xydef@\xykparsetips@{%
 \def\xyknotTIPS@@{}\def\xyknotPLACE{}%
 \let\xykparser@=\xykparsetips@@
 \xyFN@\xykparser@ }

\xydef@\xykparsetips@@{%
 \ifx\space@\next \expandafter\DN@\space{\xyFN@\xykparsetips@@}%
 \else\addLT@\ifx\next
  \ifx\xyknotTIPS@@\empty \def\xyknotTIPS@@{TF}%
  \else\DN@{FF}\ifx\next@\xyknotTIPS@@ \def\xyknotTIPS@@{TF}%
  \else\DN@{FT}\ifx\next@\xyknotTIPS@@ \def\xyknotTIPS@@{TT}%
  \fi\fi\fi
  \addLT@\DN@{\xyFN@\xykparser@}%
 \else\addGT@\ifx\next
  \ifx\xyknotTIPS@@\empty \def\xyknotTIPS@@{FT}%
  \else\DN@{FF}\ifx\next@\xyknotTIPS@@ \def\xyknotTIPS@@{FT}%
  \else\DN@{TF}\ifx\next@\xyknotTIPS@@ \def\xyknotTIPS@@{TT}%
  \fi\fi\fi
  \addGT@\DN@{\xyFN@\xykparser@}%
 \else\addEQ@\ifx\next \def\xyknotTIPS@@{TT}%
  \addEQ@\DN@{\xyFN@\xykparser@}%
 \else \ifx!\next \def\xyknotTIPS@@{FF}\DN@!{\xyFN@\xykparser@}%
 \else\DN@{\xyFN@\xykparser@}%
 \fi\fi\fi\fi\fi \next@ }
\DOCMODE)

This reads the setting for tips on string ends.

\DOCMODE(
\xydef@\xykcheckTIPS@{%
 \ifx\xyknotTIPS@@\empty \edef\xyknotTIPS@@{\xyknotTIPS@}\fi
      \DN@{FF}\ifx\next@\xyknotTIPS@@\def\tmp@{}%
 \else\DN@{FT}\ifx\next@\xyknotTIPS@@
  \expandafter\def\expandafter\tmp@\expandafter{\xykforetips@}%
 \else\DN@{TF}\ifx\next@\xyknotTIPS@@
  \expandafter\def\expandafter\tmp@\expandafter{\xykbacktips@}%
 \else 
  \DN@##1\next@{\DN@{\xykforetips@##1}}%
   \expandafter\next@\xykbacktips@\next@
  \DNii@{\expandafter\def\expandafter\tmp@\expandafter}%
   \expandafter\nextii@\expandafter{\next@}%
 \fi\fi\fi \def\xyknotTIPS@@{}}

\xydef@\xyknottips@{%
 \ifx\tmp@\empty\addtotoks@{{}}\else
  \expandafter\addtotoks@\expandafter{\expandafter{\tmp@}}\fi}
\DOCMODE)

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\end{notes}

\subsection*{Joins}
\note??=[join]
The ``joins'' are used to connect the loose ends of crossing strings.
In particular ``loops'' and ``caps'' are for placing on ends of 
horizontal or vertical `twist' and `cross' crossings, leaving the
current <pos> fixed. The ``bends'' join non-adjacent crossings of
the same type, either horizontal or vertical.

The |\xcap..| pieces are designed to join adjacent |\xover..| pieces;
they move $c$ either vertically or horizontally, 
as appropriate. Finally the |\xbend..| pieces allow for smooth joins
of $45^\circ$ slopes to horizontal or vertical slopes. For these the
actual positioning of the piece, see figure~??[f.knotjoins], 
is not entirely obvious.

\begin{figure*}[ht]
$$
\knotholesize{5pt}\UseComputerModernTips
\xygraph{!{0;/r2pc/:}="O" %[]!{\NoPSspecials}
  [ll] !{*=0\dir{o},{\vloop<>|>>><{x}|{y}>{z}},*=0\dir{x}}
   [rr]!{*=0\dir{o},{\vcap<>|>>><{x}|{y}>{z}},*=0\dir{x}}  
  [urr]!{*=0\dir{o},{\vcap-<>|>>><{x}|{y}>{z}},*=0\dir{x}}  
  [u(.5)rr]!{*=0\dir{o},{\vloop-<>|>>><{x}|{y}>{z}},*=0\dir{x}}
 [d(.5)rr]!{*=0\dir{o},{\hloop<>|>>><{x}|{y}>{z}},*=0\dir{x}}
   [rr]!{*=0\dir{o},{\hcap<>|>>><{x}|{y}>{z}},*=0\dir{x}}
  [rrr]!{*=0\dir{o},{\hcap-<>|>>><{x}|{y}>{z}},*=0\dir{x}}
  [rr]!{*=0\dir{o},{\hloop-<>|>>><{x}|{y}>{z}},*=0\dir{x}}
 "O"!{+(.5,-.5)}
 [ll]!{*\txt\small\tt{\string\vloop}} [rr]!{*\txt\small\tt{\string\vcap}} 
 [rr]!{*\txt\small\tt{\string\vcap-}}[rr]!{*\txt\small\tt{\string\vloop-}} 
 [rr]!{*\txt\small\tt{\string\hloop}}[rr]!{*\txt\small\tt{\string\hcap}} 
 [rr]!{*\txt\small\tt{\string\hcap-}}[rr]!{*\txt\small\tt{\string\hloop-}}
 "O"!{-(1,3)} %[]!{\NoPSspecials}
   [ur]!{*=0\dir{o},{\xcapv<>|>>><{x}|{y}>{z}},*=0\dir{+}} 
  [urr]!{*=0\dir{o},{\sbendv<>|>>><{x}|{y}>{z}},*=0\dir{+}}
   [r] !{*=0\dir{o},{\zbendv<>|>>><{x}|{y}>{z}},*=0\dir{+}}
  [dr] !{*=0\dir{o},{\sbendh<>|>>><{x}|{y}>{z}},*=0\dir{+}}
   [r] !{*=0\dir{o},{\zbendh<>|>>><{x}|{y}>{z}},*=0\dir{+}}
   [r] !{*=0\dir{o},{\xcaph<>|>>><{x}|{y}>{z}},*=0\dir{+}}
 "O"!{-(1.5,3.5)}
 [rr]!{*\txt\small\tt{\string\xcapv}} 
 [rr]!{*\txt\small\tt{\string\sbendv}} 
 [rr]!{*\txt\small\tt{\string\zbendv}}
 [rr]!{*\txt\small\tt{\string\sbendh}} 
 [rr]!{*\txt\small\tt{\string\zbendh}}
 [rr]!{*\txt\small\tt{\string\xcaph}} 
"O"!{+(0,-5)} %[]!{\NoPSspecials}
  [ll] !{*=0\dir{o},{\xbendr<>|>>><{x}|{y}>{z}},*=0\dir{+}}
  [ur] !{*=0\dir{o},{\xbendl<>|>>><{x}|{y}>{z}},*=0\dir{+}}
  [urr]!{*=0\dir{o},{\xbendu<>|>>><{x}|{y}>{z}},*=0\dir{+}}
  [r] !{*=0\dir{o},{\xbendd<>|>>><{x}|{y}>{z}},*=0\dir{+}}
 "O"!{-(3.5,6.5)}
 [rr]!{*\txt\small\tt{\string\xbendr}} 
 [rr]!{*\txt\small\tt{\string\xbendl}}
 [rr]!{*\txt\small\tt{\string\xbendu}} 
 [rr]!{*\txt\small\tt{\string\xbendd}}
 "O"!{+(2,-5)} %[]!{\NoPSspecials}
  [r(4)] !{*=0\dir{o},{\xbendd-<>|>>><{x}|{y}>{z}},*=0\dir{+}}
  [ur] !{*=0\dir{o},{\xbendu-<>|>>><{x}|{y}>{z}},*=0\dir{+}}
  [r]  !{*=0\dir{o},{\xbendl-<>|>>><{x}|{y}>{z}},*=0\dir{+}}
 [urr] !{*=0\dir{o},{\xbendr-<>|>>><{x}|{y}>{z}},*=0\dir{+}}
 "O"!{+(4.5,-6.5)}
 [rr]!{*\txt\small\tt{\string\xbendd-}} 
 [rr]!{*\txt\small\tt{\string\xbendu-}}
 [rr]!{*\txt\small\tt{\string\xbendl-}} 
 [rr]!{*\txt\small\tt{\string\xbendr-}}
}$$
\caption{knot joins, with orientations, labels and shifts}%
 ??=[f.knotjoins]
\end{figure*}

\noindent
Figure~??[f.knotjoins] displays the orientation on the joins. 
Also indicated are default positions for labels and arrow-tips; 
each piece uses the same code, 
e.g. |\vloop|\ |<>||>>><{x}||{y}>{z}|. Furthermore the current <pos> 
before the piece is drawn is marked using $\;^{_{\xybox{*\dir{o}}}}\;$; 
that afterwards is indicated by 
$\;^{_{\xybox{*\dir{x}}}}\;$ or $\;^{_{\xybox{*\dir{+}}}}\;$.

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The ability to scale in size and place arrow-tips, breaks, labels etc.
apply also to <join> pieces. The only difference is\dots

\note??=[join places]
The three places referred to by |<|\,,||||\,,|>| are all on a single
string. In particular |||| is always at the middle of the <join>,
whereas |<| and |>| are at {\em earlier} and {\em later} parameter
values respectively. Any adjustments??^[adjustment] involving breaks 
should occur in increasing parameter order. 

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection*{Parsing}

\DOCMODE(
\xydef@\xykparsejoin@{%
 \def\xykSCALE@@{}%
 \edef\xyknotPLACE{\xykmidPLACE@}%
 \let\xykparser@=\xykparsejoin@@
 \def\xykdefaultbreak@{\let\xykbreak@=\xykforetemp@
  \def\xyknotPLACE{(.5)}\appendtoholder@}%
%  \edef\xyknotPLACE{\xykmidPLACE@}\appendtoholder@}%
 \xyFN@\xykparsescale@ }
\DOCMODE)

\DOCMODE(
{\xyuncatcodes \catcode`@=11
\gdef\next{%
 \ifx\space@\next \expandafter\DN@\space{\xyFN@\xykparsejoin@@}%
 \else\addLT@\ifx\next \let\xykbreak@=\xykbacktemp@
  \def\xykdefaultbreak@{\def\xyknotPLACE{(.5)}%
%  \def\xykdefaultbreak@{\edef\xyknotPLACE{\xykmidPLACE@}%
  \appendtoholder@}\addLT@\DN@{\xyprecross@}%
 \else\addGT@\ifx\next \let\xykbreak@=\xykforetemp@
  \def\xykdefaultbreak@{\def\xyknotPLACE{(.5)}%
%  \def\xykdefaultbreak@{\edef\xyknotPLACE{\xykmidPLACE@}%
  \appendtoholder@}\addGT@\DN@{\xypostcross@}%
 \else\ifx|\next \let\xykbreak@=\xyknulltemp@
% \else\ifx|\next \let\xykbreak@=\xykholetemp@
  \def\xykdefaultbreak@{\def\xyknotPLACE{(.5)}%
%  \def\xykdefaultbreak@{\edef\xyknotPLACE{\xykmidPLACE@}%
  \appendtoholer@}\DN@|{\xymidcross@}%
 \else\addEQ@\ifx\next \addEQ@\DN@{\xyFN@\xykparsetips@@}%
% \else\addAT@\ifx\next \addAT@\DN@{\xykjoincontrol@}%
 \else\addAT@\ifx\next \addAT@\DN@(##1){%
  \xyadjustJOIN@i##1@@\xyFN@\xykparsejoin@@}%
 \else\DN@{\afterknot@}%
 \fi\fi\fi\fi\fi\fi \next@ }}
\xylet@\xykparsejoin@@=\next
\DOCMODE)

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\note??=[join control]
A parameter can be altered, using |@|<adjust>, to effect subtle 
adjustments to the shape of any join. Within a rectangular basis the
horizontal or vertical tangents are preserved and overall reflection
or rotation symmetry is preserved. Thus this parameter affects the
`flatness' of a cap or loop, or the amount of curvature is s-bends and
z-bends. For |\xcap..|s and |\xbend..|s the $45^\circ$ angle is altered;
this is especially useful to match the tangents when a knot-piece 
has been specified using the technique of note~??[alter size and shape].

The normal range for these parameters is between 0 and 1. Other values
can be used with interesting results---the parameter determines the 
location of control points for a B\'ezier cubic curve.

\begin{tabular}{ccl}
piece & value & effect on\dots \\
\noalign{\hrule}
|\..cap|   & .25 & flatness of cap;\\
|\..loop|  & .75 & flatness of loop;\\
|\sbend..| & .75 & curvature in the `s';\\
|\zbend..| & .75 & curvature in the `z';\\
|\xcap..|  & .5  & height of cap, slope at base;\\
|\xbend..| & .5  & curvature, slope at base.\\
\noalign{\hrule}
\end{tabular}

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{notes}

The following example gives three ways of specifying a `trefoil' knot,
using the |poly| feature to establish the location of the vertices for
knot-pieces. In each the <crossing>s are calculated to fit together
smoothly; a different way of creating <join>s is used in each.
Also the third displays subtle changes of the ??^![join control].

\begin{code}
\def\TrefoilA{\xygraph{!{0;/r.75pc/:}
 !P3"a"{~>{}}!P9"b"{~:{(1.3288,0):}~>{}}
 !P3"c"{~:{(2.5,0):}~>{}}
 !{\vover~{"b2"}{"b1"}{"a1"}{"a3"}}
 !{"b4";"b2"**\crv{"c1"}}
 !{\vover~{"b5"}{"b4"}{"a2"}{"a1"}}
 !{"b7";"b5"**\crv{"c2"}}
 !{\vover~{"b8"}{"b7"}{"a3"}{"a2"}}
 !{"b1";"b8"**\crv{"c3"}}}}
%
\def\TrefoilB{\xygraph{!{0;/r.75pc/:}
 !P3"a"{~>{}}!P9"b"{~:{(1.3288,0):}~>{}}
 !P3"c"{~:{(2.5,0):}~>{}}
 !{\vover~{"b2"}{"b1"}{"a1"}{"a3"}}
 !{\vcap~{"c1"}{"c1"}{"b4"}{"b2"}@(+.1)}
 !{\vover~{"b5"}{"b4"}{"a2"}{"a1"}}
 !{\vcap~{"c2"}{"c2"}{"b7"}{"b5"}@(+.2)}
 !{\vover~{"b8"}{"b7"}{"a3"}{"a2"}}
 !{\vcap~{"c3"}{"c3"}{"b1"}{"b8"}}}}
%
\def\TrefoilC{\xygraph{!{0;/r.75pc/:}
 !P3"a"{~>{}}
 !P12"b"{~:{(1.414,0):}~>{}}
 !{\vover~{"b2"}{"b1"}{"a1"}{"a3"}}
 !{\save 0;"b2"-"b5":"b5",
   \xcaph @(+.1)\restore}
 !{\vover~{"b6"}{"b5"}{"a2"}{"a1"}}
 !{\save 0;"b6"-"b9":"b9",
   \xcaph @(+.2)\restore}
 !{\vover~{"b10"}{"b9"}{"a3"}{"a2"}}
 !{\save 0;"b10"-"b1":"b1",
    \xcaph @(+.3)\restore} }}
$$\TrefoilA\quad\TrefoilB
   \quad\TrefoilC$$
\end{code}
\docode
\displaycode

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\DOCMODE(
\xydef@\xykjoincontrol@@{.75}%
\xydef@\xyk@joincontrol@#1{\DN@{#1}%
 \ifx\next@\empty\edef\xykjoincontrol@@{.75}%
 \else\dimen@=#1\p@
  \edef\xykjoincontrol@@{\expandafter\removePT@\the\dimen@}\fi}
\xylet@\xykjoincontrol\empty %\xyk@joincontrol@

\xydef@\xykjoincontrol@(#1){\DN@{#1}%
 \ifx\next@\empty\edef\xykjoincontrol{\xykjoincontrol@@}%
 \else\dimen@=#1\p@
  \edef\xykjoincontrol{\expandafter\removePT@\the\dimen@}%
 \fi \xyFN@\xykparsejoin@@ }

\xydef@\xykjoincontrol@i#1{\DN@{#1}%
 \ifx\next@\empty\edef\xykjoincontrol{\xykjoincontrol@@}%
 \else\dimen@=#1\p@
  \edef\xykjoincontrol{\expandafter\removePT@\the\dimen@}\fi}
\DOCMODE)

These macros are common to most joins.

\DOCMODE(
\xydef@\xykjoin@#1{%
 \expandafter\toks@\expandafter{\expandafter#1\knotSTYLE}}

\xydef@\xyksetjoin@{\xykcheckTIPS@ 
 \xykprebreak@ \xykmidbreak@ \xykpostbreak@ \xyknottips@ }
\DOCMODE)

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection*{``cap'' joins:}

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The initialisation\dots

\DOCMODE(
\xydef@\hcap{\begingroup\def\afterknot@{\xyhcap}%
 \def\xykprePLACE@{(.385)}\def\xykpostPLACE@{(.615)}%
 \def\xykmidPLACE@{(.5)}\edef\xykjoincontrol{.25}%
 \xyFN@\xykparsejoin@ }%
\xydef@\vcap{\begingroup\def\afterknot@{\xyvcap}%
 \def\xykprePLACE@{(.385)}\def\xykpostPLACE@{(.615)}%
 \def\xykmidPLACE@{(.5)}\edef\xykjoincontrol{.25}%
 \xyFN@\xykparsejoin@ }%
\xydef@\xcapv{\begingroup\def\afterknot@{\xyxcapv}%
 \def\xykprePLACE@{(.2)}\def\xykpostPLACE@{(.8)}%
 \def\xykmidPLACE@{(.5)}\edef\xykjoincontrol{.5}%
 \xyFN@\xykparsejoin@ }%
\xydef@\xcaph{\begingroup\def\afterknot@{\xyxcaph}%
 \def\xykprePLACE@{(.2)}\def\xykpostPLACE@{(.8)}%
 \def\xykmidPLACE@{(.5)}\edef\xykjoincontrol{.5}%
 \xyFN@\xykparsejoin@ }%
\DOCMODE)

The interface\dots

\DOCMODE(
\xydef@\xyvcap{\xykjoin@\xykcapv \xyksetjoin@ \xykhobject@{}}
\xydef@\xyhcap{\xykjoin@\xykcaph \xyksetjoin@ \xykvobject@{}}
\xydef@\xyxcapv{\xykjoin@\xykxcapv \xyksetjoin@ \xykvobject@{+(0,-1)}}
\xydef@\xyxcaph{\xykjoin@\xykxcaph \xyksetjoin@ \xykhobject@{+(1,0)}}
\DOCMODE)

The drawing code\dots
  
\DOCMODE(
{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7{\save,
 "_>",{\xyksaveEdgec},{\xykz_\ar #1@'{"^>"**{}?(\xykjoincontrol)@+
 ,?-(.25,0)@+,"_<";"^<"**{}?+(.25,0)@+,?(\xykjoincontrol)@+}
 "_<"#3#4#5#6},\restore \POS#7}}
\xylet@\xykcapv=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7{\save,
 "_<",{\xyksaveEdgec},{\xykz_\ar #1@'{"_>"**{}?(\xykjoincontrol)@+,
  ?+(0,.25)@+,"^<";"^>"**{}?-(0,.25)@+,?(\xykjoincontrol)@+}
 "^<"#3#4#5#6}
 \restore \POS#7}}
\xylet@\xykcaph=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7{\save,
 "^>";"_>"**{}?="^_","^<";"_<"**{}?;"^_",**{}?(\xykjoincontrol)="^_",
 "_<",{\xyksaveEdgec},{\xykz_\ar #1@'{"^_"@+}
  "^<"#3#4#5#6},\restore \POS#7}}
\xylet@\xykxcapv=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7{\save,
 "^<";"^>"**{}?="^_","_<";"_>"**{}?;"^_",**{}?(\xykjoincontrol)="^_",
 "_>",{\xyksaveEdgec},{\xykz_\ar #1@'{"^_"@+}
  "_<"#3#4#5#6},\restore \POS#7}}
\xylet@\xykxcaph=\next
\DOCMODE)

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection*{``loop'' joins:}

The initialisation and interface\dots

\DOCMODE(
\xydef@\hloop{\begingroup\def\afterknot@{\xyhloop}%
 \def\xykprePLACE@{(.07)}\def\xykpostPLACE@{(.93)}%
 \def\xykmidPLACE@{(.5)}\edef\xykjoincontrol{.75}%
 \xyFN@\xykparsejoin@ }%
\xydef@\vloop{\begingroup\def\afterknot@{\xyvloop}%
 \def\xykprePLACE@{(.07)}\def\xykpostPLACE@{(.93)}%
 \def\xykmidPLACE@{(.5)}\edef\xykjoincontrol{.75}%
 \xyFN@\xykparsejoin@ }%

\xydef@\xyvloop{\xykjoin@\xykloopv \xyksetjoin@ \xykhobject@{}}
\xydef@\xyhloop{\xykjoin@\xyklooph \xyksetjoin@ \xykvobject@{}}
\DOCMODE)

The drawing code\dots

\DOCMODE(
{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7{\save,
 "_>",{\xykz_\ar #1@'{"^>"**{}?(\xykjoincontrol)@+,?(1.25)-(.25,0)@+,
  "_<";"^<"**{}?(1.25)+(.25,0)@+,?(\xykjoincontrol)@+}"_<"#3#4#5#6}
 \restore \POS#7}}
\xylet@\xykloopv=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7{\save,
 "_<",{\xykz_\ar #1@'{"_>"**{}?(\xykjoincontrol)@+,?(1.25)+(0,.25)@+,
  "^<";"^>"**{}?(1.25)+(0,-.25)@+,?(\xykjoincontrol)@+}"^<"#3#4#5#6}
 \restore \POS#7}}
\xylet@\xyklooph=\next
\DOCMODE)

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection*{``bend'' joins:}

The initialisation\dots

\DOCMODE(
\xydef@\zbendh{\begingroup\def\afterknot@{\xyzbendh}%
 \def\xykprePLACE@{(.25)}\def\xykpostPLACE@{(.75)}%
 \def\xykmidPLACE@{(.5)}\edef\xykjoincontrol{\xykjoincontrol@@}%
 \xyFN@\xykparsejoin@ }%
\xydef@\sbendv{\begingroup\def\afterknot@{\xysbendv}%
 \def\xykprePLACE@{(.25)}\def\xykpostPLACE@{(.75)}%
 \def\xykmidPLACE@{(.5)}\edef\xykjoincontrol{\xykjoincontrol@@}%
 \xyFN@\xykparsejoin@ }%
\xydef@\sbendh{\begingroup\def\afterknot@{\xysbendh}%
 \def\xykprePLACE@{(.25)}\def\xykpostPLACE@{(.75)}%
 \def\xykmidPLACE@{(.5)}\edef\xykjoincontrol{\xykjoincontrol@@}%
 \xyFN@\xykparsejoin@ }%
\xydef@\zbendv{\begingroup\def\afterknot@{\xyzbendv}%
 \def\xykprePLACE@{(.25)}\def\xykpostPLACE@{(.75)}%
 \def\xykmidPLACE@{(.5)}\edef\xykjoincontrol{\xykjoincontrol@@}%
 \xyFN@\xykparsejoin@ }%


\xydef@\xbendr{\begingroup\def\afterknot@{\xyxbendr}%
 \def\xykprePLACE@{(.15)}\def\xykpostPLACE@{(.75)}%
 \def\xykmidPLACE@{(.5)}\edef\xykjoincontrol{.5}%
 \xyFN@\xykparsejoin@ }%
\xydef@\xbendl{\begingroup\def\afterknot@{\xyxbendl}%
 \def\xykprePLACE@{(.15)}\def\xykpostPLACE@{(.75)}%
 \def\xykmidPLACE@{(.5)}\edef\xykjoincontrol{.5}%
 \xyFN@\xykparsejoin@ }%
\xydef@\xbendu{\begingroup\def\afterknot@{\xyxbendu}%
 \def\xykprePLACE@{(.15)}\def\xykpostPLACE@{(.75)}%
 \def\xykmidPLACE@{(.5)}\edef\xykjoincontrol{.5}%
 \xyFN@\xykparsejoin@ }%
\xydef@\xbendd{\begingroup\def\afterknot@{\xyxbendd}%
 \def\xykprePLACE@{(.15)}\def\xykpostPLACE@{(.75)}%
 \def\xykmidPLACE@{(.5)}\edef\xykjoincontrol{.5}%
 \xyFN@\xykparsejoin@ }%
\DOCMODE)


The interface\dots

\DOCMODE(
\xydef@\xyzbendh{\xykjoin@\xykzbendh \xyksetjoin@ \xykhvobject@{+(1,-1)}}
\xydef@\xysbendv{\xykjoin@\xyksbendv \xyksetjoin@ \xykhvobject@{+(1,-1)}}
\xydef@\xysbendh{\xykjoin@\xyksbendh \xyksetjoin@ \xykhuobject@{+(1,1)}}
\xydef@\xyzbendv{\xykjoin@\xykzbendv \xyksetjoin@ \xykhuobject@{+(1,1)}}

\xydef@\xyxbendr{\xykjoin@\xykxbendr \xyksetjoin@ \xykhvobject@{+(1,-1)}}
\xydef@\xyxbendl{\xykjoin@\xykxbendl \xyksetjoin@ \xykhvobject@{+(0,-1)}}
\xydef@\xyxbendu{\xykjoin@\xykxbendu \xyksetjoin@ \xykhvobject@{+(1,0)}}
\xydef@\xyxbendd{\xykjoin@\xykxbendd \xyksetjoin@ \xykhvobject@{+(1,-1)}}
\DOCMODE)


The implementation\dots

\DOCMODE(
{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7{\save,
 "^<",{\xyksaveEdgec},
 {\xykz^\ar #1@'{"^>"**{}?(\xykjoincontrol)@+,
  "_>";"_<"**{}?(\xykjoincontrol)@+}"_>"#3#4#5#6}%
 \restore \POS#7}}
\xylet@\xykzbendh=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7{\save,
 "^<",{\xyksaveEdgec},
 {\xykz_\ar #1@'{"_<"**{}?(\xykjoincontrol)@+,
  "_>";"^>"**{}?(\xykjoincontrol)@+}"_>"#3#4#5#6}%
 \restore \POS#7}}
\xylet@\xyksbendv=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7{\save,
 "_<",{\xyksaveEdgec},
 {\xykz_\ar #1@'{"_>"**{}?(\xykjoincontrol)@+,
  "^>";"^<"**{}?(\xykjoincontrol)@+}"^>"#3#4#5#6}%
 \restore \POS#7}}
\xylet@\xyksbendh=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7{\save,
 "_<",{\xyksaveEdgec},
 {\xykz^\ar #1@'{"^<"**{}?(\xykjoincontrol)@+,
  "^>";"_>"**{}?(\xykjoincontrol)@+}"^>"#3#4#5#6}%
 \restore \POS#7}}
\xylet@\xykzbendv=\next


{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7{\save,
 "^<"-(0,.5),{\xyksaveEdgec},
 {\xykz_\ar #1@'{"_<";"_>"**{}?(\xykjoincontrol)@+,}"_>"#3#4#5#6}%
 \restore \POS#7}}
\xylet@\xykxbendr=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7{\save,
 "^>"-(0,.5),{\xyksaveEdgec},
 {\xykz^\ar #1@'{"_>";"_<"**{}?(\xykjoincontrol)@+,}"_<"#3#4#5#6}%
 \restore \POS#7}}
\xylet@\xykxbendl=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7{\save,
 "_<"+(.5,0),{\xyksaveEdgec},
 {\xykz_\ar #1@'{"_>";"^>"**{}?(\xykjoincontrol)@+,}"^>"#3#4#5#6}%
 \restore \POS#7}}
\xylet@\xykxbendu=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7{\save,
 "^<"+(.5,0),{\xyksaveEdgec},
 {\xykz^\ar #1@'{"^>";"_>"**{}?(\xykjoincontrol)@+,}"_>"#3#4#5#6}%
 \restore \POS#7}}
\xylet@\xykxbendd=\next

\DOCMODE)

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection*{Changing the string-style}
%
It is not necessary to use solid curves; any style available to curves
and arrows can be chosen using\dots
%
\begin{defs}
|\knotstyle{|<char>|}| &  use |\dir{|<char>|}| \cr
|\knotstyles{|<char>|}{|<char>|}| & two styles \cr
|\knotSTYLE{|<code>|}| &  use <code> \cr
\end{defs}
%
In each case the new style applies to {\em all} subsequent knot pieces,
except that the two styles apply only to crossings.
The latter case allows use of object <modifier>s. The <code> consists of
two groups |{..}{..}|\,, each containing <arrow> forms, as in
??g[xyarrow.doc:f.arrow] and notes~??g[xyarrow.doc:build arrow], 
??g[xyarrow.doc:object <modifier>s].
Only the first <arrow> form is used with <join>s whereas the two forms
are used respectively with the two strings of a <crossing> in the order
that they are drawn.

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\DOCMODE(
{\xyuncatcodes \gdef\next{{@{-}}{@{-}}}}
\xylet@\knotSTYLE=\next 

{\xyuncatcodes \gdef\next#1{\def\knotSTYLE{{@{#1}}{@{#1}}}}}
\xylet@\knotstyle=\next 

{\xyuncatcodes \gdef\next#1#2{\def\knotSTYLE{{@{#1}}{@{#2}}}}}
\xylet@\knotstyles=\next 

{\xyuncatcodes \gdef\next#1{\def\knotSTYLE{#1}}}
\xylet@\KNOTstyle=\next 
\DOCMODE)

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\paragraph*{Initialisation:}\ default values for the methods. 

\DOCMODE(
\xydef@\knotholesize@#1{\def\xykholesize{#1}}
\xydef@\xykholesize{5pt}
\xylet@\knotholesize=\knotholesize@
\xylet@\holesize=\knotholesize@

\xydef@\knottips#1{\def\xyknotTIPS@{#1}}
\xydef@\xyknotTIPS@{FF}
\xydef@\xyknotTIPS@@{}

\xydef@\xykSCALE@@{}

%\xydef@\xyknotbreak@@{}%
\xydef@\xykprebreak@@{}%
\xydef@\xykpostbreak@@{}%
\xydef@\xykmidbreak@@{}%

\xydef@\xykprePLACE@{(.25)}%
\xydef@\xykpostPLACE@{(.75)}%
\xydef@\xykmidPLACE@{}%
\xylet@\xykPLACE@=\xykmidPLACE@
\DOCMODE)

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\subsection*{Flexible crossings}

%Xtwist

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\DOCMODE(
{\gdef\next#1#2#3#4{%\message{#1:#2:#3:#4}%
 \save(0,0);"#2"-"#1"+"#4"-"#3":(.5,0):
 "#3"-"#1"+"#4"-"#2"::(0,.5)::}}
\xylet@\xykflexbase@=\next
\xylet@\xykflexbase=\xykflexbase@
\DOCMODE)

\DOCMODE(
{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8{\xykflexbase{#1}{#2}{#3}{#4}%
% \save(0,0);"#2"-"#1"+"#4"-"#3":(.5,0):
% "#3"-"#1"+"#4"-"#2"::(0,.5)::
 "#1",{\ar @{#5}@'{"#1"+(0,.125)@++(.1875,.125)@+,
 "#4"-(.1875,.25)@++(.1875,.125)@+}"#4" #6},
 "#2",{\ar @{#5}@'{"#2"+(0,.125)@++(-.1875,.125)@+,
 "#3"+(.1875,-.25)@++(-.1875,.125)@+}"#3" #7}%
\restore \POS#8}}
\xylet@\Xtwist=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8{\xykflexbase{#1}{#2}{#3}{#4}%
% \save(0,0);"#2"-"#1"+"#4"-"#3":(.5,0):
% "#3"-"#1"+"#4"-"#2"::(0,.5)::
 "#1",{\ar @{#5}@'{"#1"+(0,.0625)@++(.1875,.0625)@+,
 "#3"+(.1875,-.125)@++(-.1875,.0625)@+}"#3" #6},
 "#2",{\ar @{#5}@'{"#2"+(0,.0625)@++(-.1875,.0625)@+,
 "#4"-(.1875,.125)@++(.1875,.0625)@+}"#4" #7}%
\restore \POS#8}}
\xylet@\Xuntwist=\next
\DOCMODE)


%Xover

\DOCMODE(
{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\xykflexbase{#1}{#2}{#3}{#4}%
% \save(0,0);"#2"-"#1"+"#4"-"#3":(.5,0):
% "#3"-"#1"+"#4"-"#2"::(0,.5)::
 "#1",{\ar @{#5}@'{"#1"+(.1875,.0625)@++(.1875,.1875)@+,
 "#4"+(-#9,-.125)@+}"#4" #6},
 "#2",{\ar @{#5}@'{"#2"+(-.1875,.0625)@++(-.1875,.1875)@+,
 "#3"+(#9,-.125)@+}"#3" #7}%
\restore \POS#8}}
\xylet@\Xover=\next

{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7#8#9{\xykflexbase{#1}{#2}{#3}{#4}%
% \save(0,0);"#2"-"#1"+"#4"-"#3":(.5,0):
% "#3"-"#1"+"#4"-"#2"::(0,.5)::
 "#1",{\ar @{#5}@'{"#1"+(.1875,.0625)@++(.1875,.1875)@+,
 "#2"+(-.1875,.0625)@+}"#2" #6},
% "#3"+(#9,-.125)@+}"#3" #6},
 "#3",{\ar @{#5}@'{"#3"+(#9,-.125)@+,"#3"+(.25,-.25)@+,
% "#2",{\ar @{#5}@'{"#2"+(-.1875,.0625)@++(-.1875,.1875)@+,
 "#4"+(-#9,-.125)@+}"#4" #7}%
\restore \POS#8}}
\xylet@\Xunover=\next
\DOCMODE)


%Xcap

\DOCMODE(
{\xyuncatcodes \gdef\next#1#2#3#4#5#6#7{\xykflexbase{#1}{#2}{#3}{#4}%
% \save(0,0);"#2"-"#1"+"#3"-"#4":(.5,0):
% "#4"-"#1"+"#3"-"#2"::(0,.5)::
 "#1",{\ar @{#5}@'{"#1"+(0,.125)@++(.125,.25)@+,
 "#4"+(-.125,.375)@++(.125,-.25)@+}"#4" #6},
\restore \POS#7}}
\xylet@\Xcap=\next
\DOCMODE)


%XXover

\DOCMODE(
{\xyuncatcodes \catcode`@=11
 \gdef\next#1#2#3#4#5#6#7#8#9{\dimen@=#1\p@
 \ifdim\dimen@=\z@ \DN@{\Xunover{#2}{#3}{#4}{#5}{#8}{#6}{#7}{"#5"}}%
 \else\ifdim\dimen@>\z@
  \DNii@{#6}\ifx\nextii@\empty
    \DN@{\Xover{#2}{#3}{#4}{#5}{#8}{|(.45)\knothole}{#7}{"#5"}}%
   \else \DN@{\Xover{#2}{#3}{#4}{#5}{#8}{#6}{#7}{"#5"}}\fi
 \else
  \DNii@{#7}\ifx\nextii@\empty
   \DN@{\Xover{#2}{#3}{#4}{#5}{#8}{#6}{|(.45)\knothole}{"#5"}}%
  \else\DN@{\Xover{#2}{#3}{#4}{#5}{#8}{#6}{#7}{"#5"}}\fi
 \fi\fi 
 \def\tmp@{{#9}}\ifx\tmp@\empty\def\tmp@{{0}}\else\dimen@=#9\p@\fi
 \expandafter\next@\tmp@ }}
\xylet@\XXover=\next

{\xyuncatcodes \catcode`@=11
 \gdef\next#1#2#3#4#5#6#7#8{\dimen@=#1\p@
 \ifdim\dimen@=\z@ \DN@{\Xuntwist{#2}{#4}{#3}{#5}{#8}{#6}{#7}{"#5"}}%
 \else\ifdim\dimen@>\z@
  \DNii@{#6}\ifx\nextii@\empty
    \DN@{\Xtwist{#2}{#4}{#3}{#5}{#8}{|(.45)\knothole}{#7}{"#5"}}%
   \else \DN@{\Xtwist{#2}{#4}{#3}{#5}{#8}{#6}{#7}{"#5"}}\fi
 \else
  \DNii@{#7}\ifx\nextii@\empty
   \DN@{\Xtwist{#2}{#4}{#3}{#5}{#8}{#6}{|(.45)\knothole}{"#5"}}%
  \else\DN@{\Xtwist{#2}{#4}{#3}{#5}{#8}{#6}{#7}{"#5"}}\fi
 \fi\fi \next@ }}
\xylet@\XXtwist=\next

\DOCMODE)

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


  simplified switch-like interface


\DOCMODE(
{\xyuncatcodes \catcode`@=11
 \gdef\next#1#2#3#4{\dimen@=#1\p@
 \ifdim\dimen@=\z@ \DN@{\xykuncrossv{#4}{#2}{#3}{+(0,-1)}}%
 \else\ifdim\dimen@>\z@
  \DNii@{#2}\ifx\nextii@\empty
    \DN@{\xyktwistv{#4}{|\knothole}{#3}{+(1,0)}}%
   \else \DN@{\xyktwistv{#4}{#2}{#3}{+(1,0)}}\fi
 \else
  \DNii@{#3}\ifx\nextii@\empty
   \DN@{\xyktwistv{#4}{#2}{|\knothole}{+(0,-1)}}%
  \else\DN@{\xyktwistv{#4}{#2}{#3}{+(0,-1)}}\fi
 \fi\fi \next@ }}
\xylet@\Vcross=\next

{\xyuncatcodes \catcode`@=11
 \gdef\next#1#2#3#4{\dimen@=#1\p@
 \ifdim\dimen@=\z@ \DN@{\xykuncrossh{#4}{#2}{#3}{+(0,-1)}}%
 \else\ifdim\dimen@<\z@
  \DNii@{#2}\ifx\nextii@\empty
    \DN@{\xyktwisth{#4}{|\knothole}{#3}{+(1,0)}}%
   \else \DN@{\xyktwisth{#4}{#2}{#3}{+(1,0)}}\fi
 \else
  \DNii@{#3}\ifx\nextii@\empty
   \DN@{\xyktwisth{#4}{}{|\knothole}{+(1,0)}}%
  \else\DN@{\xyktwisth{#4}{#2}{#3}{+(1,0)}}\fi
 \fi\fi \next@ }}
\xylet@\Hcross=\next

{\xyuncatcodes \catcode`@=11
 \gdef\next#1#2#3{\dimen@=#1\p@ \save
 \ifdim\dimen@<\z@
  \DN@{\POS="@",p="@@",(0,0);(-1,0):(0,-1)::"@@";"@",}%
 \else\DN@{}\fi
 \next@ \xykcaph{#3}{#2}\restore }}
\xylet@\Hcap=\next

{\xyuncatcodes \catcode`@=11
 \gdef\next#1#2#3{\dimen@=#1\p@ \save
 \ifdim\dimen@<\z@
  \DN@{\POS="@",p="@@",(0,0);(0,-1)::"@@";"@",}%
 \else\DN@{}\fi
 \next@ \xykcapv{#3}{#2}\restore }}
\xylet@\Vcap=\next
\DOCMODE)

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\DOCMODE2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\paragraph*{The end \& Log}\leavevmode

\DOCMODE3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\DOCMODE(
\xyendinput

% $Log: xyknot.doc,v $
% Revision 3.4  1997/05/18 01:13:24  ross
% Essential bugfixes.
%
% Revision 3.3  1996/12/18  09:20:49  ross
% no changes
%
% Revision 3.1  1995/09/05 20:36:33  ross
% Release!
%
% Revision 3.0  1995/07/07  20:13:19  ross
% Major release w/new User's Guide!
%
% Revision 2.13  1995/07/04  15:04:51  ross
% Ready for release of v3.
%
% Created by Ross Moore, September 1994.
\DOCMODE)