Power sums and Homfly skein theory
Hugh R. Morton
The Murphy operators in the Hecke algebra $H_n$ of type $A$ are explicit commuting elements, whose symmetric functions are central in $H_n$. In \cite{Murphy} I defined geometrically a homomorphism from the Homfly skein $\CC$ of the annulus to the centre of each algebra $H_n$, and found an element $P_m$ in $\CC$, independent of $n$, whose image, up to an explicit linear combination with the identity of $H_n$, is the $m$th power sum of the Murphy operators. The aim of this paper is to give simple geometric representatives for the elements $P_m$, and to discuss their role in a similar construction for central elements of an extended family of algebras $H_{n,p}$.
boxedeps.tex
  %%
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  %%
 %%%%%               boxedeps.tex 
 %%%%% MACROS FOR FIGURE INSERTS OF EPSF NORM  %%%%%
 %%%%%   (EPSF = Encapsulated PostScript File)
  %%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %%  
 %%%  AUTHOR: Laurent Siebenmann
  %%    lcs@topo.math.u-psud.fr
  %%  
 %%%  VERSIONS: Feb 1991 -- Nov 1995
  %%  Revisions bi-annual approx.
  %%  
 %%%  SOMMAIRE: boxedeps.tex d\'efinit des macro-commandes
  %%    qui permettent d'int\'egrer dans un document TeX des 
  %%    objets graphiques d\'ecrits par fichier de norme EPSF,
  %%    tout en accordant a chacun le statut d'une bo\^ite TeX ayant 
  %%    les bonnes dimensions.  La (seule!) contribution unique 
  %%    de ce fichier est de faire cela d'une fa{\c}con universelle.
  %%    C'est a dire de fa{\c}con \`a pouvoir commod\'ement 
  %%    servir avec tout pilote d'imprimante de norme 
  %%    PostScript --- malgr\'e l'absence d'une norme 
  %%    pour \special. 
  %%  
 %%%% ACTIVATION:
  %%  by a driver-by-driver protocol
  %%  see \SetTexturesEPSFSpecial 
  %%  and its companions below.
  %%  
 %%%% DOCUMENTATION:
  %%  --- see boxedeps.doc
  %%  The complete utility is called BoxedEPSF but for 
  %%  portability the nickname boxedeps is useful. 
  %%
 %%%%  POSTINGS: anonymous ftp 
  %%  ---  ftp matups.math.u-psud.fr [192.54.146.4] 
  %%  is the master copy in 1995.
  %%  ---  ftp.uni-stuttgart.de 
  %%  ---  ftp ftp.TeX.AC.UK
  %%  ---  ftp ftp.SHSU.edu
  %%  Locate the full package on these three 
  %%  "CTAN" servers by using (while connected) the command
  %%        QUOTE SITE INDEX boxedeps

 %%%%  RECENT ADDITIONS: \SetepsfEPSFSpecial simultaneously serves
   %%  dvips, the Y&Y drivers, DirectTeX, CMacTeX, OzTeX>=1.7
   %%  \SetOldOzTeXEPSFSpecial  %% OzTeX 1.42--1.6x 

 
 \chardef\EPSFCatAt\the\catcode`\@
 \catcode`\@=11

 \chardef\C@tColon\the\catcode`\:
 \chardef\C@tSemicolon\the\catcode`\;
 \chardef\C@tQmark\the\catcode`\?
 \chardef\C@tEmark\the\catcode`\!
 \chardef\C@tDqt\the\catcode`\"

 \def\PunctOther@{\catcode`\:=12
   \catcode`\;=12 \catcode`\?=12 \catcode`\!=12 \catcode`\"=12}
 \PunctOther@

 %%%% MESSAGES (separate macro best for Europe)
  %%  
  \def\ms@g{\immediate\write16}

 %%%% Prevent double input
 \let\EX@\expandafter
 
 \ifx\BoxedArtundefined\relax
   \ms@g{}\ms@g{ !!!  Bad news.
     \noexpand\BoxedEPSF or \noexpand\BoxedArt macros already input.}%
   \ms@g{ !!! Hit RETURN to exit, or  c  and RETURN to forge bravely on.}%
   {\endlinechar=-1\gdef\Temp{c}%
   \global\read16 to \temp}%
     \ifx\temp\Temp 
     \else
       \EX@\EX@\EX@\end
     \fi
 \else
    \global\let\BoxedArtundefined\relax
 \fi


 
 %%temporarily suppress Plain's logging of allocations
 \let\wlog@ld\wlog 
 \def\wlog#1{\relax} 

 %% New for TOOLS
 %\newif\ifIN@
 \newdimen\XShift@ \newdimen\YShift@ 
 \newtoks\Realtoks
 
 %%% New for BoxedEPSF
 %
 \newdimen\Wd@ \newdimen\Ht@
 \newdimen\Wd@@ \newdimen\Ht@@
 %
 \newdimen\TT@
 \newdimen\LT@
 \newdimen\BT@
 \newdimen\RT@
 %
 \newdimen\XSlide@ \newdimen\YSlide@ 
 %
 \newdimen\TheScale  %% secretly scale in mils: 1pt= 1mil 
 \newdimen\FigScale  %% secretly scale in mils: 1pt= 1mil 
 %
 \newdimen\ForcedDim@@

 \newtoks\EPSFDirectorytoks@
 \newtoks\EPSFNametoks@
 \newtoks\BdBoxtoks@
 \newtoks\LLXtoks@  %% useful info for Oz
 \newtoks\LLYtoks@  
 \newtoks\URXtoks@  %% useful for epsf.tex syntax
 \newtoks\URYtoks@  

  
 \newif\ifRepeat@
 \newif\ifNotIn@
 \newif\ifForcedDim@
 \newif\ifForceOn@
 \newif\ifForcedHeight@
 \newif\ifPSOrigin
 \newif\ifPCfmt@
 \newif\ifPCfmtOK@


 \newread\EPSFile@ 
 \newwrite \epsf@log
 \def\WrLog@{\immediate\write\epsf@log}

 \gdef\InitLog@{%
   \def\epsf@log@name{epsf.lst}%
   \count255=\time\divide\count255 by 60
   \edef\temp@{\the\count255}%
   \multiply\count255 by -60\relax 
   \advance\count255 by \time
   \immediate\openout\epsf@log \epsf@log@name
   \WrLog@{ \pct@@\space boxedeps.tex log, date 
   \the\day-\the\month-\the\year, 
   time \temp@:\the\count255.}\WrLog@{}%
   \ms@g{}\ms@g{ **** See file \epsf@log@name\space
    for list of EPSFs integrated by boxedeps.tex}%
   \global\let\InitLog@\relax}

 \def\noArtList{\global\let\InitLog@\relax}
 \let\noEPSFList\noArtList
 \def\noArtReports{\global\let\ReportEPSF@\relax}
 \let\noEPSFReports\noArtReports
 
 %%%% WORD-PROCESSING MACROS
  %%
  %%% \IN@0#1@#2@ : Is 1st exp of #1 in 1st exp of #2 ??
   %% Answer in \ifIN@
 \newif\ifIN@\def\IN@{\EX@\INN@\EX@}
  \long\def\INN@0#1@#2@{\long\def\NI@##1#1##2##3\ENDNI@
    {\ifx\m@rker##2\IN@false\else\IN@true\fi}%
     \EX@\NI@#2@@#1\m@rker\ENDNI@}
  \def\m@rker{\m@@rker}

  %%%  \SPLIT@0#1@#2@  :  Split 1st exp of #2 at 1st exp of #1
   %%  \Initialtoks@ , \Terminaltoks@ will contain pieces
  \newtoks\Initialtoks@  \newtoks\Terminaltoks@
  \def\SPLIT@{\EX@\SPLITT@\EX@}
  \def\SPLITT@0#1@#2@{\def\TTILPS@##1#1##2@{%
     \Initialtoks@{##1}\Terminaltoks@{##2}}\EX@\TTILPS@#2@}

 %%%% MACROS TO TRIM  \ForeTrim@0#1@ and \Trim@0#1@  
   %% result appears in \Trimtoks@
   %% LIMITATION: assume no multiple spaces to trim

  \newtoks\Trimtoks@

  %%% \ForeTrim@0#1@ trims initial space of first erpansion of #1
   %% #1 of form \the\toks0 or \mymacro
 \def\ForeTrim@{\EX@\ForeTrim@@\EX@}
 \def\ForePrim@0 #1@{\Trimtoks@{#1}}
 \def\ForeTrim@@0#1@{\IN@0\m@rker. @\m@rker.#1@%
     \ifIN@\ForePrim@0#1@%
     \else\Trimtoks@\EX@{#1}\fi}
   %%\m@rker expands here to \m@@rker since spot initial,
   %% so no confusuion with \m@rker

  %%% \Trim@0#1@ trims init and terminal spaces 
   %% Same syntax.
   %% Warns if internal spaces found.
   %% 
  \def\Trim@0#1@{%
    \ForeTrim@0#1@%
    \IN@0 @\the\Trimtoks@ @%
    \ifIN@ 
      \SPLIT@0 @\the\Trimtoks@ @\Trimtoks@\Initialtoks@
      \IN@0\the\Terminaltoks@ @ @%
      \ifIN@ \else \Trimtoks@{FigNameWithSpace}\fi
    \fi }


  %%%% MATH MACROS (provisional)
    %% use dimen registers for reals; unit 1pt
    %% (numerical dimension arguments OK unless contrary noted)

  %%%% One needs the point token seq (pt with cat 12) USES dimen 0
   \newtoks\pt@ks
   \def\getpt@ks 0.0#1@{\pt@ks{#1}}
   \dimen0=0pt\relax\EX@\getpt@ks\the\dimen0@

   %%% Convert dimen to "decimal multiplier"% USES dimens 0,2
  %\newtoks\Realtoks% the output!
  \def\Real#1{%
    \dimen2=#1%
      \SPLIT@0\the\pt@ks @\the\dimen2@%%  lop off the points
       \Realtoks=\Initialtoks@ %\showthe\Realtoks
            }

   %%% Multiplication 
      % USES dimens 0,2,4,6; preserves args; output \Product
   \newdimen\Product
   \def\Mult#1#2{%
     \dimen4=#1\relax
     \dimen6=#2%
     \Real{\dimen4}%
     \Product=\the\Realtoks\dimen6%
        }

   %%% Inverse 
     % USES dimens 0; preserves arg; output \Inverse
 \newdimen\Inverse
 \newdimen\hmxdim@ \hmxdim@=8192pt%halfmaxdimen
 \def\Invert#1{%
  \Inverse=\hmxdim@
  \dimen0=#1%
  \divide\Inverse \dimen0%
  \multiply\Inverse 8}

 %%% \Rescale#1#2#3  % USES dimens 0,2,4,6
  %%  alters dimen register #1 by ratio #2/#3 
  %%  where #2,#3 can be raw dimensions OR dimen registers
   \def\Rescale#1#2#3{% Adequate accuracy. Can improve. 
              \divide #1 by 100\relax
              \dimen2=#3\divide\dimen2 by 100 \Invert{\dimen2}% 
              \Mult{#1}{#2}%
              \Mult\Product\Inverse 
              #1=\Product}

 %%% \Scale#1 scales dimen register #1 
  %%  by dimen register real \TheScale; USES dimen 0
  \def\Scale#1{\dimen0=\TheScale 
      \divide #1 by  1280 %% 1280*5120*10=1000*2^16 
      \divide \dimen0 by 5120  
      \multiply#1 by \dimen0 
      \divide#1 by 10   %% max size of #1 about 32000/10 pt
     }
 
 %%% SCRUNCHING BOXES AND SHIFTING CONTENTS
  %% TeX has to do this in general
  %% since some drivers do not let 
  %% one do it readily using Postscript

 \newbox\scrunchbox

 %%% \Scrunched#1 puts #1 in an hbox
  %%    then in effect zeros the dimensions of this box
 \def\Scrunched#1{{\setbox\scrunchbox\hbox{#1}%
   \wd\scrunchbox=0pt
   \ht\scrunchbox=0pt
   \dp\scrunchbox=0pt
   \box\scrunchbox}}

  %%% \Shifted@#1 puts #1 in \hbox 
   %% then locates basepoint to bottom left corner
   %% then translates ink only by \XShift@,\YShift@
   %% with Postscript convention
   %% For simplicity use only on scrunched boxes
  %\newdimen\XShift@ 
  %\newdimen\YShift@ 
 \def\Shifted@#1{%
   \vbox {\kern-\YShift@
       \hbox {\kern\XShift@\hbox{#1}\kern-\XShift@}%
           \kern\YShift@}}

  %%% \cBoxedEPSF#1 the main macro
   %%  component macros are explained in order below

 \def\cBoxedEPSF#1{{\leavevmode 
    %% double brace for amstex \allign, \alligned, ...
   \ReadNameAndScale@{#1}%
   \SetEPSFSpec@
   \InitLog@ 
   \ReadEPSFile@ \ReadBdB@x  
   %% Calculations
     \TrimFigDims@ 
     \CalculateFigScale@  
     \ScaleFigDims@
     \SetInkShift@
   \hbox{$\mathsurround=0pt\relax
         \vcenter{\hbox{%
             \FrameSpider{\hskip-.4pt\vrule}%
             \vbox to \Ht@{\offinterlineskip\parindent=\z@%
                \FrameSpider{\vskip-.4pt\hrule}\vfil 
                \hbox to \Wd@{\hfil}%
                \vfil
                \InkShift@{\EPSFSpecial{\EPSFSpec@}{\FigSc@leReal}}%
             \FrameSpider{\hrule\vskip-.4pt}}%
         \FrameSpider{\vrule\hskip-.4pt}}}%
     $\relax}%
    \CleanRegisters@ 
    \ReportEPSF@ 
    }}
 
 \def\tBoxedEPSF#1{\setbox4\hbox{\cBoxedEPSF{#1}}%
     \setbox4\hbox{\raise -\ht4 \hbox{\box4}}%
     \box4
      }

 \def\bBoxedEPSF#1{\setbox4\hbox{\cBoxedEPSF{#1}}%
     \setbox4\hbox{\raise \dp4 \hbox{\box4}}%
     \box4
      }

  \let\BoxedEPSF\cBoxedEPSF% default setting

  %% Some compatibility with BoxedArt.tex
   %
   \let\BoxedArt\BoxedEPSF

  %% Some compatibility with Sweet-teX
   %
  \def\gLinefigure[#1scaled#2]_#3{%
        \BoxedEPSF{#3 scaled #2}}
    
  %% Some compatibility with Rokicki's dvips
   %
  \let\EPSFbox\bBoxedEPSF \let\EPSFfile\bBoxedEPSF
  
  \def\EPSFxsize{\afterassignment\ForceW@\ForcedDim@@}
      \def\ForceW@{\ForcedDim@true\ForcedHeight@false}
  
  \def\EPSFysize{\afterassignment\ForceH@\ForcedDim@@}
      \def\ForceH@{\ForcedDim@true\ForcedHeight@true}

  \def\EmulateRokicki{%
       \let\epsfbox\bBoxedEPSF \let\epsffile\bBoxedEPSF
       \let\epsfxsize\EPSFxsize \let\epsfysize\EPSFysize} 
 
 %%% \ReadNameAndScale@#1
  %
 \def\ReadNameAndScale@#1{\IN@0 scaled@#1@% DOUBLE BARRELED
   \ifIN@\ReadNameAndScale@@0#1@%
   \else \ReadNameAndScale@@0#1 scaled\DefaultMilScale @%
   \fi}
  
 \def\ReadNameAndScale@@0#1scaled#2@{% HELPER MACRO
    \let\OldBackslash@\\%
    \def\\{\OtherB@ckslash}%
    \edef\temp@{#1}%
    \Trim@0\temp@ @%
    \EPSFNametoks@\EX@{\the\Trimtoks@ }%
    \FigScale=#2 pt%
    \let\\\OldBackslash@
    }
 
 \def\SetDefaultEPSFScale#1{%
      \global\def\DefaultMilScale{#1}}

 \SetDefaultEPSFScale{1000}


 %%% \ReadEPSFile@
  %
 \def \SetBogusBbox@{%
     \global\BdBoxtoks@{ BoundingBox:0 0 100 100 }%
     \global\def\BdBoxLine@{ BoundingBox:0 0 100 100 }%
     \ms@g{ !!! Will use placeholder !!!}%
     }

 {\catcode`\%=12\relax\gdef\P@S@{%!}\gdef\pct@@{%%}}
   %% %! min sign of PS file

 %%% Opening and closing
 \def\ReadEPSFile@{%\show\EPSFSpec@%
   \openin\EPSFile@\EPSFSpec@
   \relax  %necessary to prevent precocious expansion of \ifeof
  \ifeof\EPSFile@
   \ms@g{}%
   \ms@g{ !!! EPS FILE \the\EPSFDirectorytoks@
       \the\EPSFNametoks@\space WAS NOT FOUND !!!}%
   \SetBogusBbox@
  \else
   \ReadEPSFile@@
  \fi
  \closein\EPSFile@ 
   }
 
 %% \AdvanceLcnt@ for PC EPSF case only
 \global\let\AdvanceLcnt@\relax %% initialize
 \def\AdvanceLcnt@@{%
   \advance\count255 1 \ifnum \count255>50 \Lcnt@ms@g \fi}
  
 \def\Lcnt@ms@g{\ms@g{}%
  \ms@g{ !!! PC binary EPS preview probably before BoundingBox.}%
  \ms@g{ !!! To accellerate, use utility to move all previews after \pct@@ EOF.}%
  \global\let\AdvanceLcnt@\relax
  \global\let\AdvanceLcnt@@\relax %% in all, one test, one complaint
  }

 %%% Header tests
 \def\ReadEPSFile@@{%
   \begingroup%%
   \chardef\other=12 \chardef\cmt=14 \catcode"0C=9 \catcode"7F=\cmt 
   \def\do##1{\catcode`##1=\other}\dospecials \catcode`\ =10
   \catcode"00=\cmt \catcode`\\=\cmt \catcode`\!=12 \catcode`\:=12 
   %% next to recognize PC-type header
   \global\PCfmt@false%% initialize
   \catcode"C5=\other \catcode"D0=\other \catcode"D3=\other \catcode"C6=\cmt 
   \global\read\EPSFile@ to \HdLine@ 
   \IN@0\P@S@ @\HdLine@ @%
   \ifIN@ %% %! accepted as %!PS so do BdBox search loop!!
     %\message{*********}%
     %\showthe\catcode`\%
     %\showthe\catcode`\!
     %\show\HdLine@
     %\show\P@S@
     \let\temp@\ReadEPSFile@@@
   \else
     \def\temp@@{^^c5^^d0^^d3}%
     \ifx\temp@@\HdLine@  %% sign of PC binary PostScript file header
       \global\PCfmt@true
       \count255=0 %% will be line counter
       \let\AdvanceLcnt@\AdvanceLcnt@@
       \let\temp@\ReadEPSFile@@@
       \ifPCfmtOK@\else %%
         \ms@g{}%
         \ms@g{ !!! PC binary EPS file header encountered. Designated }%
         \ms@g{ !!! driver may not accept this and similar EPS files.}%
         \relax\global\PCfmtOK@true %% just one warning
       \fi
     \else
       \ms@g{}%
       \ms@g{ !!! \the\EPSFNametoks@\space is not PostScript.}%
       \ms@g{ !!! It should begin with the "\P@S@" or PC binary header.}%
       \SetBogusBbox@
     \fi
   \fi
   \temp@
  \endgroup%\relax
  }

 %%% BoundingBox search loop
 \def\ReadEPSFile@@@{\Repeat@true
   \loop   
   \ifeof\EPSFile@\relax\Repeat@false 
     \ms@g{}%
     \ms@g{ !!! BoundingBox NOT FOUND IN %
       \the\EPSFDirectorytoks@\the\EPSFNametoks@\space!!! }%
     \SetBogusBbox@
   \else
     \ReadEPSFile@@@@
   \fi
   \ifRepeat@
   \AdvanceLcnt@ %% is \relax unless PC EPSF; >50 => complaint
   \repeat
   \global\BdBoxtoks@\expandafter{\BdBoxLine@}%
   }

 %%% BoundingBox or HiResBoundingBox
  %% HiRes normally on line following if present at all
 \def\ReadEPSFile@@@@{%
   \global\read\EPSFile@ to \BdBoxLine@
   %\BdBoxtoks@\EX@{\BdBoxLine@}\ms@g{}\ms@g{ *  \the\BdBoxtoks@}%%%%
   \IN@0BoundingBox@\BdBoxLine@ @%
   \ifIN@
     %\ms@g{}\ms@g{ *  SCORE!!}
     \Repeat@false
     \read\EPSFile@ to \temp@
     \IN@0HiResBoundingBox:@\temp@ @%
     \ifIN@
       \global\let\BdBoxLine@\temp@
     \fi
   \fi}

  %%% \ReadBdB@x
   % Rmk For simplicity 0 not used in syntax 
   %  of \ReadBdB@x@,  \ReadBdB@x@@ 
  \def\ReadBdB@x{% PART 0
   \EX@\ReadBdB@x@\the\BdBoxtoks@ @}
  
  \def\ReadBdB@x@#1BoundingBox:#2@{% PART 1
    \ForeTrim@0#2@%
    \IN@0atend@\the\Trimtoks@ @%
       \ifIN@ \Trimtoks@={0 0 100 100 }%
         \ms@g{}%
         \ms@g{ !!! BoundingBox not found in %
         \the\EPSFDirectorytoks@\the\EPSFNametoks@\space !!!}%
         \ms@g{ !!! It must not be at end of EPSF !!!}%
         \ms@g{ !!! Will use placeholder !!!}%
       \fi%% cf \SetBogusBbox@
    \EX@\ReadBdB@x@@\the\Trimtoks@ @%
   }
    
  \def\ReadBdB@x@@#1 #2 #3 #4@{% PART 2
      \Wd@=#3bp\advance\Wd@ by -#1bp%
      \Ht@=#4bp\advance\Ht@ by-#2bp%
       \Wd@@=\Wd@ \Ht@@=\Ht@ %% useful info for Clark
       \LLXtoks@={#1}\LLYtoks@={#2}%% useful for Oz
       \URXtoks@={#3}\URYtoks@={#4}%% useful for epsf.tex
      \ifPSOrigin\XShift@=-#1bp\YShift@=-#2bp\fi 
     }

  %%% \SetEPSFDirectory 
   %
   \def\G@bbl@#1{}
   \bgroup
     \global\edef\OtherB@ckslash{\EX@\G@bbl@\string\\}
   \egroup

  \def\SetEPSFDirectory{%  Part 1
           \bgroup\PunctOther@\relax
           \let\\\OtherB@ckslash
           \SetEPSFDirectory@}

 \def\SetEPSFDirectory@#1{% Part 2
    \edef\temp@{#1}%
    \Trim@0\temp@ @%  result in \Trimtoks@
    \global\toks1\EX@{\the\Trimtoks@ }\relax
    \egroup
    \EPSFDirectorytoks@=\toks1
    }

 \def\SetEPSFSpec@{%
     \bgroup
     \let\\=\OtherB@ckslash
     \global\edef\EPSFSpec@{%
        \the\EPSFDirectorytoks@\the\EPSFNametoks@}%
     \global\edef\EPSFSpec@{\EPSFSpec@}%
     \egroup}

 %%% Is trimming done percentage-wise??
 \newif\ifPCT@ 

 \def\PCTtrueErrTst@{}%
 \def\PCTfalseErrTst@{}%

 %%%% Trimming figure dimensions (source)
   %%
  %%% \TrimSide@ 
   %% #1 is one of \TT@,...,\RT@;  #2 is arg of \TrimTop,...
 \def\TrimSide@#1#2{%
    \IN@0pct@#2@%
    \ifIN@
       \PCT@true\PCTtrueErrTst@
       \SPLIT@0pct@#2@%
       \advance#1 by \the\Initialtoks@ pt
    \else
       \PCTfalseErrTst@
       \advance#1 by #2
    \fi}
 
 \def\TrimTop#1{\TrimSide@{\TT@}{#1}}%
 \def\TrimLeft#1{\TrimSide@{\LT@}{#1}}%
 \def\TrimBottom#1{\TrimSide@{\BT@}{#1}}%
 \def\TrimRight#1{\TrimSide@{\RT@}{#1}}%

 \def\TrimBoundingBox#1{%
   \TrimTop{#1}%
   \TrimLeft{#1}%
   \TrimBottom{#1}%
   \TrimRight{#1}%
       }

 %%% Convert percentage trim to dimension trim on source (if nec)
  %% #1 is percent trim (pt is unit); #2 is source width or height
 \def\PCTtoSideTrimDims@{%
    \ifPCT@
      \Rescale\LT@\Wd@{100pt}%\showthe\LT@
      \Rescale\RT@\Wd@{100pt}%
      \Rescale\TT@\Ht@{100pt}%
      \Rescale\BT@\Ht@{100pt}%
    \fi}

 \def\TrimFigDims@{%
    \PCTtoSideTrimDims@
    \advance\Wd@ by -\LT@ 
    \advance\Wd@ by -\RT@ \RT@=\z@
    \advance\Ht@ by -\TT@ \TT@=\z@
    \advance\Ht@ by -\BT@ 
    }
  

 %%% \CalculateFigScale@
  %
  \def\ForceWidth#1{\ForcedDim@true
       \ForcedDim@@#1\ForcedHeight@false}
  
  \def\ForceHeight#1{\ForcedDim@true
       \ForcedDim@@=#1\ForcedHeight@true}

  \def\ForceOn{\ForceOn@true}
  \def\ForceOff{\ForceOn@false\ForcedDim@false}
  
  \def\CalculateFigScale@{%
            %Have default \FigScale or read \FigScale
     \ifForcedDim@\FigScale=1000pt% %% start afresh
           \ifForcedHeight@
                \Rescale\FigScale\ForcedDim@@\Ht@
           \else
                \Rescale\FigScale\ForcedDim@@\Wd@
           \fi
     \fi
     \Real{\FigScale}%
     \edef\FigSc@leReal{\the\Realtoks}%
     }
   
  \def\ScaleFigDims@{\TheScale=\FigScale
      \ifForcedDim@
           \ifForcedHeight@ \Ht@=\ForcedDim@@  \Scale\Wd@
           \else \Wd@=\ForcedDim@@ \Scale\Ht@
           \fi
      \else \Scale\Wd@\Scale\Ht@        
      \fi
      \ifForceOn@\relax\else\global\ForcedDim@false\fi
      \Scale\LT@\Scale\BT@  %%%\Scale\Wd@\Scale\Ht@
      \Scale\XShift@\Scale\YShift@
      }
      
  %%% \ShowDisplacementBoxes
   %%  shows (prints) corrected scaled and positioned
   %%  bounding boxes; for diagnostics
  %%% \HideDisplacementBoxes makes them invisible again
   %%
 \def\HideDisplacementBoxes{\global\def\FrameSpider##1{\null}}
 \def\ShowDisplacementBoxes{\global\def\FrameSpider##1{##1}}
 \let\HideFigureFrames\HideDisplacementBoxes %% some synonyms
 \let\ShowFigureFrames\ShowDisplacementBoxes
 \ShowDisplacementBoxes
 
  %%% \hSlide#1, \vSlide#1
   %%
 \def\hSlide#1{\advance\XSlide@ by #1}
 \def\vSlide#1{\advance\YSlide@ by #1}
 
  %%% \SetInkShift@, \InkShift@#1
   %%
  \def\SetInkShift@{%
            \advance\XShift@ by -\LT@
            \advance\XShift@ by \XSlide@
            \advance\YShift@ by -\BT@
            \advance\YShift@ by -\YSlide@
             }
   %
  \def\InkShift@#1{\Shifted@{\Scrunched{#1}}}
 
  %%% \CleanRegisters@
   %
  \def\CleanRegisters@{%
      \globaldefs=1\relax
        \XShift@=\z@\YShift@=\z@\XSlide@=\z@\YSlide@=\z@
        \TT@=\z@\LT@=\z@\BT@=\z@\RT@=\z@
      \globaldefs=0
      \global\PCT@false}

 \def\ReportEPSF@{%
  \ms@g{ *** Box composed for the EPS file \the\EPSFNametoks@}%
  \edef\temp@{\the\EPSFDirectorytoks@}%
  \ifx\temp@\LastEPSFDirectory@\else
    \WrLog@{--- change EPSF directory ==> \temp@}%
    \global\let\LastEPSFDirectory@\temp@
  \fi
  \def\temp@{}%
  \ifPCfmt@ \edef\temp@{\space \pct@@\space EPSP binary}%
  \fi
  %\show\temp@
  \WrLog@\EX@{\the\EPSFNametoks@\temp@}%
  }

 \gdef\LastEPSFDirectory@{}%% initialize

 %%% Special syntax for several drivers. The macros 
  %% \SetTexturesEPSFSpecial  %% Textures 
  %% \SetUnixCoopEPSFSpecial %% dvi2ps early unix 
  %% \SetNiponUnixEPSFSpecial %% dvi2ps j-version 
  %% \SetBechtolsheimDVI2PSEPSFSpecial and 
  %% \SetBechtolsheimDVITPSEPSFSpecial %% by S.P.Bechtolsheim
  %% \SetLisEPSFSpecial %% dvi2ps by Tony Lis
  %% \SetepsfEPSFSpecial %% ******* will work for many:
  %% ----- dvips, the Y&Y drivers, DirectTeX, CMacTeX, OzTeX>=1.7
  %% \SetRokickiEPSFSpecial %% dvips by Tom Rokicki
  %% \SetDVIPSoneEPSFSpecial %% DVIPSONE of Y&Y same as
  %% \SetDVIWindowEPSFSpecial %% DVIWindow of Y&Y 
  %% \SetDirectTeXEPSFSpecial %% DirectTeX by Wilfried Ricken
  %% \SetCMacTeXEPSFSpecial %% CMacTeX by Tom Kiffe
  %% \SetOzTeXEPSFSpecial %% OzTeX (>=1.7) by Andrew Trevorrow
  %% -----
  %% \SetOldOzTeXEPSFSpecial %% OzTeX 1.42--1.6x 
  %% \SetPSprintEPSFSpecial %% PSprint by Andrew Trevorrow
  %%  --- also for OzTeX versions <= 1.41 !!
  %% \SetArborEPSFSpecial %% ArborTeX DVILASER/PS
  %% \SetClarkEPSFSpecial %% dvitops by James Clark
  %% \SetBeebeEPSFSpecial %% DVIALW by N. Beebe
  %% \SetNorthlakeEPSFSpecial %% Northlake Software
  %% \SetStandardEPSFSpecial %% Nonexistant: Placebo used
  %%
  %% These many drivers are supported roughly
  %% by (re-)defining the macro \EPSFSpecial#1#2, where
  %% #1 = EPS file pathname (use \\ for the letter backslash)
  %% #2 = scale in mils 
  %% Be wary of using strange characters in pathnames!
 
 %% Textures, Blue Sky Research, Barry Smith
 \def\SetTexturesEPSFSpecial{\PSOriginfalse%\PSOrigintrue
  \gdef\EPSFSpecial##1##2{\relax
    \edef\specialtemp{##2}%
    \SPLIT@0.@\specialtemp.@\relax
    \special{illustration ##1 scaled
                        \the\Initialtoks@}}}
 
  %% Unix : dvi2ps by:  Mark Senn, Stephan  Bechtolsheim,  
   % Bob  Brown, Richard, Furuta, James Schaad, Robert  Wells, 
   % Norm Hutchinson, Neal Holt, Scott Jones, Howard Trickey.
   % Introduced by B. Horn <bkph@ai.mit.edu>
  \def\SetUnixCoopEPSFSpecial{\PSOrigintrue % Please test!
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt% convert real to dimen
      \divide\dimen4 by 1000\relax
      \Real{\dimen4}%dimens 0,2 used here
      \edef\Aux@{\the\Realtoks}%  
      %%convert dimen to real
      \special{psfile=##1\space 
        hscale=\Aux@\space
        vscale=\Aux@}}}

  %% Unix : dvi2ps j-version 1.91j
   % Introduced by Takashi Matsushita matsu@tmuds1.phys.metro-u.ac.jp
   % Takafumi Sakurai, sakurai@math.s.chiba-u.ac.jp
   % Kazuhiro Kazama,  kazama@square.ntt.jp
  \def\SetNiponUnixEPSFSpecial{\PSOriginfalse % Please test!
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt% convert real to dimen
      \divide\dimen4 by 1000\relax
      \Real{\dimen4}%dimens 0,2 used here
      \edef\Aux@{\the\Realtoks}%  
      %%convert dimen to real
      \special{epsfile=##1\space 
        hscale=\Aux@\space
        vscale=\Aux@}}}

  %% Unix : dvi2ps j-version 1.91j
   % Introduced by Takashi Matsushita matsu@tmuds1.phys.metro-u.ac.jp
   % Takafumi Sakurai, sakurai@math.s.chiba-u.ac.jp
   % Kazuhiro Kazama,  kazama@square.ntt.jp
  \def\SetNiponUnixEPSFSpecial{\PSOriginfalse % Please test!
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt% convert real to dimen
      \divide\dimen4 by 1000\relax
      \Real{\dimen4}%dimens 0,2 used here
      \edef\Aux@{\the\Realtoks}%  
      %%convert dimen to real
      {\dimen6=\the\URYtoks@ pt %
      \advance\dimen6 by -\the\LLYtoks@ pt %
      \dimen6=\Aux@\dimen6 
      \Real{\dimen6}% 
      \XShift@=0bp\YShift@=\the\Realtoks bp
      \Shifted@{\special{epsfile=##1\space 
        hscale=\Aux@\space
        vscale=\Aux@}}}}}
%%%      \Shifted@{\special{illustration ##1}}}}}



  %% dvi2ps and dvitps by S.P. Bechtolsheim,
   % Introduced by B. Horn <bkph@ai.mit.edu> and Carl.M.Jones, 
   % testing by R. Evans <Robert@cm.cardiff.ac.uk>
   % and George Denk <denk@mathematik.tu-muenchen.de>
   % Note that a prolog file psfig.pro
   % specific to the driver should be available.
  \def\SetBechtolsheimEPSFSpecial@{%% tool macro only
   \PSOrigintrue
   \special{\DriverTag@ Include0 "psfig.pro"}%
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt %% convert real to dimen
      \divide\dimen4 by 1000\relax
      \Real{\dimen4} %% dimens 0,2 used here
      \edef\Aux@{\the\Realtoks}%% convert dimen to real
      \special{\DriverTag@ Literal "10 10 0 0 10 10 startTexFig
           \the\mag\space 1000 div 
           dup 3.25 neg mul 1 index .25 neg mul translate %% correction line
           \Aux@\space mul dup scale "}%
      \special{\DriverTag@ Include1 "##1"}%
      \special{\DriverTag@ Literal "endTexFig "}%
        }}

  \def\SetBechtolsheimDVITPSEPSFSpecial{\def\DriverTag@{dvitps: }%
      \SetBechtolsheimEPSFSpecial@}

  \def\SetBechtolsheimDVI2PSEPSFSSpecial{\def\DriverTag@{DVI2PS: }%
      \SetBechtolsheimEPSFSpecial@}

  %% dvi2ps by Tony Lis,
   % implantations? ; dates?; availability?
   % Introduced by B. Horn <bkph@ai.mit.edu>
  \def\SetLisEPSFSpecial{\PSOrigintrue 
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt% convert real to dimen
      \divide\dimen4 by 1000\relax
      \Real{\dimen4}% dimens 0,2 used here
      \edef\Aux@{\the\Realtoks}%  
      %%convert dimen to real
      \special{pstext="10 10 0 0 10 10 startTexFig\space
           \the\mag\space 1000 div \Aux@\space mul 
           \the\mag\space 1000 div \Aux@\space mul scale"}%
      \special{psfile=##1}%
      \special{pstext=endTexFig}%
        }}

  %% dvips by Tom Rokicki; free driver in portable C 
   % Introduced by W.D. Neumann <neumann@mps.ohio-state.edu>
  \def\SetRokickiEPSFSpecial{\PSOrigintrue 
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt% convert real to dimen
      \divide\dimen4 by 10\relax
      \Real{\dimen4}% dimens 0,2 used here
      \edef\Aux@{\the\Realtoks}%  
      %%convert dimen to real
      \special{psfile="##1"\space 
        hscale=\Aux@\space
        vscale=\Aux@}}}

  \def\SetInlineRokickiEPSFSpecial{\PSOrigintrue 
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt% convert real to dimen
      \divide\dimen4 by 1000\relax
      \Real{\dimen4}% dimens 0,2 used here
      \edef\Aux@{\the\Realtoks}%  
      %%convert dimen to real
      \special{ps::[begin] 10 10 0 0 10 10 startTexFig\space
           \the\mag\space 1000 div \Aux@\space mul 
           \the\mag\space 1000 div \Aux@\space mul scale}%
      \special{ps: plotfile ##1}%
      \special{ps::[end] endTexFig}%
        }}

  %%% Rokicki's epsf.tex syntax.
   %% Is followed by DirectTeX (W. Ricken)
   %% CMacTeX (Tom Kiffe), Y&Y's drivers, and OzTeX >= 1.7
  \def\SetepsfEPSFSpecial{\PSOriginfalse 
   \gdef\epsfclipstring{}%
   %% make { clip} if you want to clip (non-portably)
   %% down to the PostScript bounding box. (OzTeX refuses this)
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt %% convert mil scale to dimen
      \divide\dimen4 by 100 %% 1000ths to 10ths
      \Real{\dimen4}%% dimens 0,2 used here
      \edef\theScaleholder@{\the\Realtoks}%
      \dimen6=\the\URXtoks@ pt %% not bp!!!!!
      \advance\dimen6 by -\the\LLXtoks@ pt
      %% \dimen6 now (bbox width) * points
      \dimen6=\theScaleholder@\dimen6 \Real{\dimen6}%% dimens 0,2 used here
      \edef\Temp{\the\Realtoks}%
      %% \Temp now holds decimal real no
      %% 10 times desired "real width" (rwi) in bp
      \dimen6=\the\URYtoks@ pt %
      \advance\dimen6 by -\the\LLYtoks@ pt %
      \dimen6=\theScaleholder@\dimen6 \Real{\dimen6}% 
      \special{PSfile=##1\space
        llx=\the\LLXtoks@\space lly=\the\LLYtoks@\space
        urx=\the\URXtoks@\space ury=\the\URYtoks@\space 
        rwi=\Temp\space rhi=\the\Realtoks
        \epsfclipstring}%
        }}
    %% Rmk: Under epsf.tex only multiples of (1/10) bp available widths
    %% but considerably more accuracy available here.

  \let\SetDirectTeXEPSFSpecial=\SetepsfEPSFSpecial
  \let\SetCMacTeXEPSFSpecial=\SetepsfEPSFSpecial
  \let\SetOzTeXEPSFSpecial=\SetepsfEPSFSpecial
 %%% \SetOzTeXEPSFSpecial only for OzTeX in versions 1.7 and later

 %%% Old OzTeX versions (1.42 to 1.6), by Andrew Trevorrow.
 %%% For versions <= 1.41 use \SetPSprintEPSFSpecial below.
 %%  Nota: Version 1.42--1.5 may give
 %%  spurious "offpage" error notices on printing.
 \def\SetOldOzTeXEPSFSpecial{\PSOrigintrue
 \gdef\EPSFSpecial##1##2{%
 \dimen4=##2pt%% convert real to dimen
 \divide\dimen4 by 1000\relax
 \Real{\dimen4}%% dimens 0,2 used here
 \edef\Aux@{\the\Realtoks}%% convert dimen to real
 \special{epsf=\string"##1\string"\space scale=\Aux@}%
 }} 

 %% PSprint, by AndrewTrevorrow for VaX VMS
 %% and OzTeX versions <= 1.41  
  % tested 2-91 by Max Calviani <ISICA@ASTRPD.infn.it>
  \def\SetPSprintEPSFSpecial{\PSOriginfalse % artifice; see below
   \gdef\EPSFSpecial##1##2{%note order
     \special{##1\space 
       ##2 1000 div \the\mag\space 1000 div mul
       ##2 1000 div \the\mag\space 1000 div mul scale
       \the\LLXtoks@\space neg \the\LLYtoks@\space neg translate
       }}}

 %% DVILASER/PS driver originally written by David Fuchs
  % marketed and supported by ArborTeXt  535 W. William St.
  % Suite 300, Ann Arbor, MI 48103, U.S.A
  % (313) 996-3566 (313) 996-3573
  % help@arbortext.com, Andrew Dobrowolski
 \def\SetArborEPSFSpecial{\PSOriginfalse % check!
   \gdef\EPSFSpecial##1##2{%
     \edef\specialthis{##2}%
     \SPLIT@0.@\specialthis.@\relax % suppress decimals (nec!)
     \special{ps: epsfile ##1\space \the\Initialtoks@}}}

 %% dvitops, (c) James Clark <jjc@jclark.uucp>
  % public domain; distributed by UK TeX Archive
  % computers: unix, msdos, vms, primos and vm/cms,
  % introduced by S. Ratz <spqr@uk.ac.southampton.ecs>
 \def\SetClarkEPSFSpecial{\PSOriginfalse % please test!
   \gdef\EPSFSpecial##1##2{%
     \Rescale {\Wd@@}{##2pt}{1000pt}%
     \Rescale {\Ht@@}{##2pt}{1000pt}%
     \special{dvitops: import 
           ##1\space\the\Wd@@\space\the\Ht@@}}}

 %% DVIPSONE, for PC compatibles
  % Y&Y, 106 Indian Hill, Carlisle MA 01741, USA
  % (508) 371-3286
  % (introduced by B. Horn <bkph@ai.mit.edu>)
  \let\SetDVIPSONEEPSFSpecial\SetepsfEPSFSpecial
  \let\SetDVIPSoneEPSFSpecial\SetepsfEPSFSpecial
  \let\SetDVIWindowEPSFSpecial\SetepsfEPSFSpecial

 %% DVIALW by N. Beebe, public domain 
  % DVI Driver Distribution, Center for Scientific Computing,
  % Department of Mathematics, 220 South Physics Building,
  % University of Utah, Salt Lake City, UT 84112, USA
  % (introduced by B. Horn <bkph@ai.mit.edu>)
  % Proposed standard; see TUGboat article 1993.
  \def\SetBeebeEPSFSpecial{%please test!
   \PSOriginfalse% 
   \gdef\EPSFSpecial##1##2{\relax
    \special{language "PS",
      literal "##2 1000 div ##2 1000 div scale",
      position = "bottom left",
      include "##1"}}}
  \let\SetDVIALWEPSFSpecial\SetBeebeEPSFSpecial

 %% Northlake software
  \def\SetNorthlakeEPSFSpecial{\PSOrigintrue
   \gdef\EPSFSpecial##1##2{%
     \edef\specialthis{##2}%
     \SPLIT@0.@\specialthis.@\relax % suppress decimals (nec!)
     \special{insert ##1,magnification=\the\Initialtoks@}}}

 %%% MultiSpecial for Textures PLUS drivers supporting epsf.tex
 %%% syntax of Rokicki (one of many dvips syntaxes)
  %% *experimental* 2-95
  \def\SetEPSFMultiSpecial{\PSOriginfalse
   \csname newwrite\endcsname \EPSFile@@
   \immediate\openout\EPSFile@@ illustration
   \immediate\write\EPSFile@@ {\P@S@}%
   \immediate\write\EPSFile@@ {\pct@@ Remark: Tranquilizer for OzTeX...}%
   \immediate\closeout\EPSFile@@
   \gdef\EPSFSpecial##1##2{%
      \dimen4=##2pt %% convert mil scale to dimen
      \divide\dimen4 by 100 %% 1000ths to 10ths
      \Real{\dimen4}%% dimens 0,2 used here
      \edef\theScaleholder@{\the\Realtoks}%
      \dimen6=\the\URXtoks@ pt %% not bp!!!!!
      \advance\dimen6 by -\the\LLXtoks@ pt
      %% \dimen6 now (bbox width) * points
      \dimen6=\theScaleholder@\dimen6 \Real{\dimen6}%% dimens 0,2 used here
      \edef\Temp{\the\Realtoks}%
      %% \Temp now holds decimal real no
      %% 10 times desired "real width" (rwi) in bp
      \dimen6=\the\URYtoks@ pt %
      \advance\dimen6 by -\the\LLYtoks@ pt %
      \dimen6=\theScaleholder@\dimen6 \Real{\dimen6}% 
      \special{PSfile=##1\space
        llx=\the\LLXtoks@\space lly=\the\LLYtoks@\space
        urx=\the\URXtoks@\space ury=\the\URYtoks@\space 
        rwi=\Temp\space rhi=\the\Realtoks}%
    \edef\specialtemp{##2}%
    \SPLIT@0.@\specialtemp.@\relax
    \special{illustration ##1 scaled \the\Initialtoks@}%
    }}

 \def\SetStandardEPSFSpecial{%
   \gdef\EPSFSpecial##1##2{%
     \ms@g{}
     \ms@g{%
       !!! Sorry! There is still no standard for \noexpand
       \special EPSF integration !!!}%
     \ms@g{%
      --- So you will have to identify your driver using a command}%
     \ms@g{%
      --- of the form \string\Set...EPSFSpecial, in order to get}%
     \ms@g{%
      --- your graphics to print.}%  
     \ms@g{%
      Update your BoxedEPSF and read the documentation.}%
     \ms@g{}
     \gdef\EPSFSpecial####1####2{}
     }}

  \SetStandardEPSFSpecial %% currently gives warning
 
 \let\wlog\wlog@ld %%restore logging 

 \catcode`\:=\C@tColon
 \catcode`\;=\C@tSemicolon
 \catcode`\?=\C@tQmark
 \catcode`\!=\C@tEmark
 \catcode`\"=\C@tDqt

 \catcode`\@=\EPSFCatAt

  \endinput

%%%%%%%%%%%% ASCII Character test
 %
 %       Upper case letters: ABCDEFGHIJKLMNOPQRSTUVWXYZ
 %       Lower case letters: abcdefghijklmnopqrstuvwxyz
 %                                   Digits: 0123456789
 % Square, curly, angle braces, parentheses: [] {} <> ()
 %           Backslash, slash, vertical bar: \ / |
 %                              Punctuation: . ? ! , : ;
 %          Underscore, hyphen, equals sign: _ - =
 %                Quotes--right left double: ' ` "
 %"at", "number" "dollar", "percent", "and": @ # $ % &
 %           "hat", "star", "plus", "tilde": ^ * + ~
 %
 %%%%%%%%%%%%%%%%%%%%%%%%
 %
 % Une seule erreur de transmission peut empoisoner un programme!
 %
 % A single transmission error can poison a whole program.
 %
 %%%%%%%%%%%%%%%%%%%%%%%%

gtmonout.tex
%%%
%%%  This is gtmonout.tex. It contains routines which automatically
%%%  generate:  
%%%  (1)  the title page in correct format
%%%  (2)  TeX file for the reprint cover (cover.tex)
%%%  (3)  a short batch file to rename the reprint cover file (covname.bat)
%%%  (4)  a data file (paper.dat) for the final publication routine
%%%  (5)  a text version (paper.txt) for email to subscribers and abstract
%%%  (6)  an html version (paper.htm) for the html abstract.
%%%
%%%  Used with makemon.bat, this completely automates the final
%%%  production of GTM articles and reprint cover, and largely automates
%%%  the final publication. 
%%%
%%%  For instructions see gtoutput.txt.
%%%                                               Colin Rourke  26.7.02

%  test for latex or plain tex
\def\ifplaintex{\expandafter\ifx\csname documentclass\endcsname\relax}

%  get print centerpage:

\ifplaintex 
\hoffset 14truemm
\voffset 31truemm
\else
\headsep 23pt
\footskip 35pt
\hoffset -4truemm
\voffset 12.5truemm
\fi

\def\gt{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\ 
$\cal T\!\!$opology}}        %  journal title in recommended style

\def\gtm{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\ 
$\cal T\!\!$opology $\cal M\mskip-1mu$onographs}}    %  for monographs

\def\gtp{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\ 
$\cal T\!\!$opology $\cal P\!$ublications}}  % GT publications

\def\Addressesr{\bigskip
{\small \parskip 0pt \leftskip 0pt \rightskip 0pt plus 1fil \def\\{\par}
\sl\theaddress\par
\medskip
\rm Email:\stdspace\tt\theemail\hfill\rm Received:\qua\receiveddate \par}}

\def\Addressesr{\bigskip
{\small \parskip 0pt \leftskip 0pt \rightskip 0pt plus 1fil \def\\{\par}
\sl\theaddress\par
\medskip
\rm Email:\stdspace\tt\theemail\hfill\rm Received:\qua\receiveddate \par}}

\def\recd{{\small Received:\qua\receiveddate\ifx\reviseddate\relax
\else\qquad Revised:\qua\reviseddate\fi\par}} 

%  define the various new ingredients of the title page and the data
%  output files

\def\lognumber#1{\def\thelognumber{#1}}
\def\volumenumber#1{\def\thevolumenumber{#1}}
\def\volumeyear#1{\def\thevolumeyear{#1}}
\def\volumename#1{\def\thevolumename{#1}}
\def\papernumber#1{\def\thepapernumber{#1}}
\def\pagenumbers#1#2{\def\startpage{#1}\def\finishpage{#2}}
\def\published#1{\def\publishdate{#1}}
\def\received#1{\def\receiveddate{#1}}
\def\revised#1{\def\reviseddate{#1}}
\def\accepted#1{\def\accepteddate{#1}}
\def\asciititle#1{\def\theasciititle{#1}}
\def\covertitle#1{\def\thecovertitle{#1}}
\def\coverauthors#1{\def\thecoverauthors{#1}}
\def\asciiauthors#1{\def\theasciiauthors{#1}}
\def\asciiaddress#1{\def\theasciiaddress{#1}}
\def\asciiemail#1{\def\theasciiemail{#1}}
\def\asciiurl#1{\def\theasciiurl{#1}}
\def\coverauthors#1{\def\thecoverauthors{#1}}
\long\def\asciiabstract#1{\long\def\theasciiabstract{#1}}
\def\asciikeywords#1{\def\theasciikeywords{#1}}
\def\erratum#1{\def\theerratum{non-empty}\def\erratumnumber{#1}}
\def\shortauthors#1{\def\theshortauthors{#1}}
\def\shorttitle#1{\def\theshorttitle{#1}}

%  initialise

\let\\\par\let\thelognumber\relax
\let\thevolumenumber\relax\let\thepapernumber\relax
\let\thevolumeyear\relax\let\startpage\relax
\let\finishpage\relax\let\publishdate\relax\let\receiveddate\relax
\let\reviseddate\relax\let\accepteddate\relax\let\theasciititle\relax
\let\thecovertitle\relax\let\theasciiauthors\relax\let\theasciiaddress\relax
\let\theasciiabstract\relax\let\theasciikeywords\relax
\let\thecoverauthors\relax
\let\thecoverauthors\relax\let\theerratum\relax\let\theasciiemail\relax
\let\theshortauthors\relax\let\theshorttitle\relax\let\theasciiurl\relax

%%% Define a few things for test purposes
\def\startpage{1}\def\finishpage{15}\def\thepapernumber{77}

%%%% Edit the next three lines for later volumes (or include definitions
%%%% with papers):
\volumenumber{2}
\volumename{Proceedings of the Kirbyfest}
\volumeyear{1999}



\long\def\maketitlep{   % start of definition of \maketitlep

\count0=\startpage

\gtm\nl        %   GT mongraphs (top left) 
{\small Volume \thevolumenumber: \thevolumename\nl 
\ifx\theerratum\relax\else Erratum \erratumnumber\nl\fi
Pages \startpage--\finishpage\nl}

\vglue 0.1truein   % top margin

% title
{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par\smallskip}{\ifplaintex\large
\else\Large\fi\bf\thetitle}\par\medskip}   
\vglue 0.05truein 

% authors :
%
{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par}{\sc\theauthors}
\par\medskip}%
 
\vglue 0.03truein 

%  abstract and classification numbers:

{\small\leftskip 25pt\rightskip 25pt{\bf Abstract}\stdspace\theabstract

{\bf AMS Classification}\stdspace\theprimaryclass
\ifx\thesecondaryclass\relax\else; \thesecondaryclass\fi\par
{\bf Keywords}\stdspace \thekeywords\par}\vglue 7pt

}   % end of definition of \maketitlep

%%% Headers and footers

\font\phead=cmsl9 scaled 950
\font\lhead=cmsl9 scaled 1050
\font\pnum=cmbx10 scaled 913
\font\lnum=cmbx10 
\font\pfoot=cmsl9 scaled 950
\font\lfoot=cmsl9 scaled 1050
\ifplaintex
\headline{\vbox to 0pt{\vskip -4.5mm\line{\small\phead\ifnum
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m4-15.tex
%%%%%%%%%%%%%%  Geometry and Topology Monogrpahs: m4-15.tex  %%%%%%%%
%%%%        
%%%%               Power sums and Homfly skein theory                     
%%%%             
%%%%                         Hugh R. Morton
%%%%                          
%%%%                  Published in Volume 4(2002) 235-244
%%%%
%%%%                    Publication date 13 October 2002
%%%%
%%%%                        This is a LaTeX file
%%%%
%%%%
%%%%%%%%%%%%%%%%%%                                   %%%%%%%%%%%%%%%%%%%
\documentclass{gtart}

\input gtmonout
\volumenumber{4}
\volumename{Invariants of knots and 3-manifolds (Kyoto 2001)}
\volumeyear{2002}
\papernumber{15}
\pagenumbers{235}{244}
\received{31 October 2001}
\revised{15 May 2002}
\accepted{22 July 2002}
\published{13 October 2002}


\usepackage[rokicki, hideboxes]{boxedeps}




\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem*{conjecture}{Conjecture}
\theoremstyle{definition}
\newtheorem*{remark}{Remark}


\newtheorem*{Def}{Definition}



\newcommand{\pic}[2]{\BoxedEPSF{#1 scaled #2}}

\def\CA{{\cal A}}
\def\CC{{\cal C}}
\def\CS{{\cal S}}


\def\Xor{\pic{Xor.ART} {400}}
\def\Yor{\pic{Yor.ART} {400}}
\def\Ior{\pic{Ior.ART} {400}}


\def\Idor{\pic{Idor.ART} {400}}
\def\Rcurlor{\pic{Rcurlor.ART} {400}}

\def\Murphy{\pic{Murphy.ART} {400}}
\def\Murphysum{\pic{Murphys.ART} {400}}

\def\Gcross{\pic{Gcross.ART} {300}}
\def\Apos{\pic{Gcross.ART} {250}}
\def\Amix{\pic{Amix.ART} {250}}
\def\Aneg{\pic{Aneg.ART} {250}}
\def\Aij{\pic{Aij.ART} {300}}


\def\Tjrev{\pic{Tjrev.ART} {300}}
\def\Unp{\pic{Unp.ART} {300}}



\def\XC{\pic{XC.ART} {400}}
\def\XCYC{\pic{XCYC.ART} {400}}
\def\YCXC{\pic{YCXC.ART} {400}}
\def\XYC{\pic{XYC.ART} {400}}
\def\TC{\pic{CloseT.ART} {400}}
\def\figthree{\pic{Fnp.ART} {400}}


\def\PsiD{\pic{PsiD.ART} {400}}
\def\PsiX{\pic{PsinpX.ART} {400}}

\def\PsiXY{\pic{PsinpXY.ART} {400}}
\def\PsiXPsiY{\pic{PsinpXPs.ART} {400}}

\newcommand\ds{\displaystyle}
\newcommand\eval[1]{\langle#1\rangle}





%


\begin{document}

\title{Power sums and Homfly skein theory}                    
\authors{Hugh R. Morton}                  
\address{Department of Mathematical Sciences, University of 
Liverpool\\Peach St, Liverpool, L69 7ZL, UK}                  
\email{morton@liv.ac.uk}
\url{http://www.liv.ac.uk/\char'176su14}


\begin{abstract}   
The Murphy operators in the Hecke algebra
$H_n$ of type $A$ are explicit commuting
elements, whose 
symmetric functions are  central in $H_n$.
In \cite{Murphy} I defined geometrically a  homomorphism
from the Homfly skein 
$\CC$ of the annulus to the centre of each algebra $H_n$, and
found an element $P_m$ in $\CC$, independent of $n$, 
whose image, up to an explicit linear combination with the identity of $H_n$,  is the $m$th power sum of the
Murphy operators. The aim of this paper is to give simple geometric representatives for the elements $P_m$, and to discuss their role in a similar construction for central elements of an extended family of algebras $H_{n,p}$.
\end{abstract}

\asciiabstract{   
The Murphy operators in the Hecke algebra H_n of type A are explicit
commuting elements, whose symmetric functions are central in H_n.  In
[Skein theory and the Murphy operators, J. Knot Theory Ramif. 11
(2002), 475-492] I defined geometrically a homomorphism from the
Homfly skein C of the annulus to the centre of each algebra H_n, and
found an element P_m in C, independent of n, whose image, up to an
explicit linear combination with the identity of H_n, is the m-th
power sum of the Murphy operators. The aim of this paper is to give
simple geometric representatives for the elements P_m, and to discuss
their role in a similar construction for central elements of an
extended family of algebras H_{n,p}.}



\primaryclass{57M25}                
\secondaryclass{20C08}              
\keywords{Homfly skein theory, Murphy operators, power sums,
supersymmetric polynomials, annulus, Hecke algebras}                    

\makeshorttitle  %%% Makes a short header for AGT and GTM articles




 
\section*{Introduction}

This article is a combination of material presented at  a conference in
Siegen in January 2001 and at the Kyoto low-dimensional topology workshop in
September 2001. I am grateful to the organisers of both these meetings for
the opportunity to discuss this work.
 I must also thank the Kyoto organisers and RIMS for their hospitality during
the production of this account.

Since much of the material in the talks is already contained in 
\cite{Murphy} I  refer readers there for many of the details, while giving a
brief review here. In this paper I  concentrate on some of the related
results and developments which did not appear in \cite{Murphy}. In particular
I  include a direct skein theory proof in theorem \ref{braidsum} of a result
originally due to Aiston \cite{Adams} giving an unexpectedly simple geometric
representation for a sequence of elements $\{P_m\}$  in the skein of the
annulus  as  sums of a small number of closed braids. The  elements $P_m$
were interpreted in \cite{Adams} as the result of applying Adams operations
to the core curve of the annulus, when the skein of the annulus is viewed as
the representation ring of $sl(N)$ for\eject 
large $N$.  The same elements $P_m$ are shown in \cite{Murphy} to
give rise to the power sums of the Murphy operators in the centre of
the Hecke algebras $H_n$.

I extend the ideas to algebras $H_{n,p}$ based on tangles with strings
allowed in both directions at top and bottom, where a similar uniform
description for much, in fact probably all,  of the centre, can be given.
This suggests analogues of the Murphy operators in these cases, which now
come in two sets $\{T(j)\}, j=1,\ldots,n$ and $\{U(k)\}, k=1,\ldots,p$, of
commuting elements, and provide central elements as the supersymmetric
functions of these sets of elements in the algebra $H_{n,p}$.
 
\section{Framed Homfly skeins}

The skein theory setting  described more fully in \cite{Murphy} is based on
the framed Homfly skein relations,
\begin{eqnarray*}
\Xor\  -\ \Yor& =&(s-s^{-1})\ \Ior\\
\mbox{and }\  \Rcurlor &=&v^{-1}\ \Idor.
\end{eqnarray*}
The  framed Homfly skein ${\cal S}(F)$ of a planar surface $F$, with some
designated input and output boundary points, is defined as linear
combinations of oriented tangles in $F$, modulo Reidemeister moves II and III
and these two local skein relations.
 The coefficient ring can be taken as $\Lambda={\bf Z}[v^{\pm1}, s^{\pm1}]$
with powers of $s^k-s^{-k}$ in the denominators.


When  $F$ is a rectangle with $n$ inputs at the bottom and
$n$ outputs at the top the skein $R_n^n(v,s)={\cal S}(F)$ of
$n$-tangles is an algebra under composition of tangles, isomorphic to the
Hecke algebra
$H_n(z)$, with $z=s-s^{-1}$ and coefficients extended to $\Lambda$.
 This algebra has a presentation with generators
$\{\sigma_i\}, i=1,\ldots,n-1$ corresponding to Artin's elementary braids,
which satisfy the braid relations
 and the quadratic relations
$\sigma_i^2=z\sigma_i+1$.
The braids $T(j), j=1,\ldots,n$, shown in figure 1 make up a set of commuting
elements in $H_n$. These are shown by Ram \cite{Ram} to be the Murphy
operators of Dipper and James \cite{Dipper}, up to linear combination with
the identity of $H_n$.
\begin{center}
$T(j)\ =\ $ \Murphy\\ {\small Figure 1}
\end{center}
Their properties are discussed further in \cite{Murphy}, where the element
$T^{(n)}$ in figure 2 is shown to represent their sum, again up to linear
combination with the identity.
\begin{center}
$T^{(n)}\ =\ $ \Murphysum\\ {\small Figure 2}
\end{center}
 In the same paper the power sums $\sum T(j)^m$ are presented as skein
elements which clearly belong to the centre of $H_n$, in terms of an element
$P_m$ in the skein $\CC$ of the annulus. This leads to skein theory
presentations for any symmetric function of the Murphy operators as central
elements of $H_n$, and gives a pictorial view of the result of Dipper and
James which identifies the centre of $H_n$ for generic parameter with these
symmetric functions.

Before giving the  skein presentation for $P_m$ as a sum of $m$ closed braids
in the annulus I discuss briefly the construction of central elements in
$H_n$, and in some extended variants of these algebras.

\begin{Def} Write $H_{n,p}$ for the skein $\CS(F)$ where $F$ is the rectangle
with $n$ outputs and $p$ inputs at the top, and matching inputs and outputs
at the bottom as in figure 3.
\end{Def}
\begin{center}
$F\ =\ $ \figthree\\ {\small Figure 3}
\end{center} There is  a natural algebra structure on $H_{n,p}$ induced by
placing oriented tangles one above the other. When $p=0$ we have the Hecke
algebra $H_n=H_{n,0}$. The resulting algebra $H_{n,p}$ has been studied by
Kosuda and Murakami, \cite{Kosuda}, in the context of $sl(N)_q$ endomorphisms
of the module $V^{\otimes n}\otimes \overline{V}^{\otimes p}$, where $V$ is
the fundamental $N$-dimensional module. Hadji \cite{Hadji} has also described
an explicit skein-theoretic basis for it; while there is a linear isomorphism
of $H_{n,p}$ with $H_{(n+p)}$ this is not in general an algebra isomorphism.

\section{The annulus} The skeins $H_{n,p}$ are closely related to the skein
$\CC=\CS(F)$ where $F$ is the annulus. An element  $X\in \CC$, which is
simply a linear combination of diagrams in the annulus, modulo the skein
relations, is indicated schematically as 
\[\XC\ .\]
The closure map $H_{n,p}\to {\cal C}$, induced by taking a tangle $T$ to its
closure $\hat {T}$ in the annulus, is defined by \[\hat {T}\ =\ \TC\ .\] This
is a $\Lambda$-linear map, whose image we  call ${\CC}_{n,p}$. Every diagram
in the annulus lies in some $\CC_{n,p}$.

The skein $\CC$ has a product induced by placing one annulus outside another, 
\[\XYC\ =\ \XCYC\ =\ \YCXC .\]
 Under this product $\CC$ becomes a commutative algebra, since diagrams in
the inner annulus can be moved across the outer annulus using Reidemeister
moves II and III.

The evaluation map $\eval{\ }:\CC\to\Lambda$ is defined by setting $\eval X$
to be the framed Homfly polynomial of $X$, regarded as a linear combination
of diagrams in the plane. When the Homfly polynomial is normalised to take
the value 1 on the empty diagram, and hence the value
$\delta=\ds\frac{v^{-1}-v}{s-s^{-1}}$ on the zero-framed unknot, the map
$\eval{\ }$ is a multiplicative homomorphism.

\section{Central elements of $H_{n,p}$} There is  an easily defined algebra 
homomorphism $\psi_{n,p}$ from
${\cal C}
$ to the centre of each algebra $H_{n,p}$, induced from
$D=\PsiD$  by placing $X\in \CC$  around the circle and the identity of
$H_{n,p}$ on the arc, to get
\[\psi_{n,p}(X)\ =\ \PsiX\ \in H_{n,p}.\]
 It is clear that 
\[\psi_{n,p}(XY)\ =\ \PsiXY\ =\ \PsiXPsiY\ =\psi_{n,p}(X)\psi_{n,p}(Y),\] and
that
 the elements $\psi_{n,p}(X)$  all lie in the centre of $H_{n,p}$. It is
shown in \cite{Murphy} that the image of $\psi_{n,0}$, called $\psi_n$ there,
consists of all symmetric polynomials in the Murphy operators in $H_n$ and
so, by \cite{Dipper}, makes up the whole of the centre of $H_n$ in the
generic case. 

It is natural to suspect that the same is true for $H_{n,p}$.
\begin{conjecture} The image of $\psi_{n,p}$ is the whole centre of $H_{n,p}$.
\end{conjecture}
\begin{remark}  I don't have an
 immediate skein theory argument showing that central elements can always be
written as $\psi(X)$, even in the case of  $H_n$.   I suspect that the best
way would be to show that the restriction of $\psi_{n,p}$ to $\CC_{n,p}$
followed by closure is an isomorphism, and couple this with an upper bound on
the dimension of the centre, although I don't know of an algebraic
determination of the centre in the general case $n,p>0$. 
\end{remark}

In \cite{Murphy} the element $P_m\in \CC$ is constructed, by manipulating
generating functions, as a polynomial in elements $\{h_i\}$, while each $h_i$
is itself initially a linear combination of $i!$ closed braids. It satisfies
\[\psi_{n,0}(P_m-\eval{P_m})=(s^m-s^{-m})v^{-m}\sum_{j=1}^n T(j)^m,\] for
every $n$. This result can be generalised to $H_{n,p}$ as follows.
\begin{theorem} The central element $\psi_{n,p}(P_m)$ of $H_{n,p}$ can be
written, up to linear combination with the identity, as the power sum
difference \[v^{-m}\sum_{j=1}^n T(j)^m-v^m\sum_{k=1}^pU(k)^m,\] for some
commuting elements $\{T(j)\}\cup\{U(k)\}$.
\end{theorem}

\begin{proof} Apply the arguments of \cite{Murphy} using the skein $\CA$ of
the annulus with a boundary input and output to show that
\[\psi_{n,p}(P_m)-\eval{P_m}{\rm Id}=(s^m-s^{-m})\left(v^{-m}\sum_{j=1}^n
T(j)^m-v^m\sum_{k=1}^p U(k)^m\right),\]  where 
\[T(j)\ =\ \Tjrev\ ,\quad U(k)\ =\ \Unp\ .\]
\end{proof} 
\begin{remark}
Every central element of the form $\psi_{n,p}(X)$ is a
polynomial in the elements $\psi_{n,p}(P_m)$. Stembridge 
\cite{Stembridge} defines a supersymmetric polynomial in two sets of
commuting variables $\{x_i\}$ and $\{y_j\}$ to be a polynomial which is
symmetric in $\{x_i\}$ and $\{y_j\}$ separately, and which is independent of
$t$ after making the substitution $x_1=y_1=t$. He then characterises them as
polynomials in the power sum differences $\sum x_i^m-\sum y_j^m$.  The
central elements
$\psi_{n,p}(X)$ can thus be described  as  supersymmetric polynomials in the
sets of commuting elements $\{v^{-1}T(j)\}$ and $\{vU(k)\}$.

These elements  might be
considered
  as a set of Murphy operators in the  general setting.  There is certainly
some measure of choice for the set of elements; conjugating all of them by a
fixed element will not alter their supersymmetric polynomials, while further
choices are also possible. 
\end{remark}



\section{Geometric representatives of $P_m$}


The main result of this paper is to give a geometrically simple 
representative for $P_m$ as a sum of  $m$ closed braids.



Write $A_m \in {\cal C}$ for the closure of the
$m$-braid \[\sigma_{m-1}\cdots \sigma_2\sigma_1\ =\ \Gcross\ .\] The central
element $T^{(n)}\in H_n$ can be written as $T^{(n)}=\psi_{n,0}(A_1)$, so we
can take
 $P_1$ to be the core curve $A_1$ of the annulus.


The construction of the elements $P_m$ in \cite{Murphy} makes use of the
sequence of elements $h_i\in \CC_{i,0}$ which arise as the closure of one of
the two most basic idempotents in $H_i$. The properties of these elements
which will be used here are their relations to the elements $A_m$. These are
expressed compactly in terms of formal power series with coefficients in the
ring $\CC$ in theorem \ref{AH}. 

\begin{theorem}[Morton] \label{AH}
 \[A(t)=\frac{H(st)}{H(s^{-1}t)},\] where
\[H(t)=1+\sum_{n=1}^\infty h_n t^n\] and
 \[A(t)=1+(s-s^{-1})\sum_{m=1}^\infty A_mt^m.\]
\end{theorem} 
\begin{proof} This is given in \cite{Murphy}, using simple skein properties
of the elements $h_n$ and the skein $\CA$ of the annulus with a single input
on one boundary curve and a matching output on the other.
\end{proof}

The elements $P_m$ are defined in \cite{Murphy} by the formula
\[\sum_{m=1}^\infty \frac{P_m}{m}t^m=\ln(H(t)).\]
Now every skein admits a {\em mirror map},
$\overline{\phantom{w}}:\CS(F)\to\CS(F)$ induced by switching all crossings
in a tangle, coupled with inverting $v$ and $s$ in $\Lambda$. When this is
applied to the series $A(t)$ the result is the series
\[\overline{A}(t)=1-z\sum_{m=1}^\infty \overline{A}_mt^m,\] where
$\overline{A}_m$ is the mirror image of the closed braid $A_m$, and
$z=s-s^{-1}$.

 The series $H(t)$ is invariant under the mirror map, as shown in
\cite{Murphy}, and hence so is $P_m$, although I will not need  these facts
here.  The simple geometric representative for $P_m$ in theorem
\ref{braidsum} can be thought of as `almost' $A_m$, but averaged in  a way
that ensures mirror symmetry.
\begin{Def} Write $A_{i,j}$ for the closure of the braid 
\[\sigma_{i+j}^{-1}\sigma_{i+j-1}^{-1}\ldots\sigma_{i+1}^{-1}\sigma_i\ldots\sigma_2\sigma_1
=\ \Aij\] on $i+j+1$ strings.
\end{Def} Then  $A_{i,j}$ is  the closed braid with $i$ positive and $j$
negative crossings which results from switching the last $j$ crossings of
$A_{i+j+1}$, with $A_m=A_{m-1,0}$ and $\overline{A}_m=A_{0,m-1}$.



\begin{theorem}[Aiston] \label{braidsum} The element $P_m$ is a multiple of
the sum of all $m$ of the closed braids $A_{i,j}$ which have $m$ strings.
Explicitly,
\[[m]P_m=\sum_{{i=0}\atop{i+j=m}}^{m-1}A_{i,j-1}=\overline{A}_m+\ldots+{A}_m
=\Aneg +\cdots +\Amix +\cdots +\Apos,\]  where $\ds
[m]=\frac{s^m-s^{-m}}{s-s^{-1}}$.
\end{theorem}

While Aiston's proof required a diversion through results about $sl(N)_q$
representations, the proof here uses only the skein-based result of theorem
\ref{AH} and one further simple skein theory lemma.

\begin{lemma}\label{Adiff} For all $i,j>0$ we have
\[A_{i,j-1}-A_{i-1,j}=zA_{i-1,0}A_{0,j-1}\ (=zA_i\overline{A}_j).\]
\end{lemma}
\begin{proof} Apply the skein relation to the $i$th positive crossing in
$A_{i,j-1}$. Switching this crossing gives $A_{i-1,j}$ while smoothing it 
gives the product $A_i\overline{A}_j$.
\end{proof}

\begin{corollary}\label{mirror}
\[A(t)\overline{A}(t)=1.\]
\end{corollary} 
\begin{proof} We have
\[
z\sum_{{i=1}\atop{i+j=m}}^{m-1}A_i\overline{A}_j=A_{m-1,0}-A_{0,m-1}=A_m-\overline{A}_m,m>1,\]
by lemma \ref{Adiff}. The coefficient of $t^m$ in $A(t)\overline{A}(t)$ is
\[zA_m-z\overline{A}_m-z^2\sum_{i+j=m}A_i\overline{A}_j =0, m>0.\]
\end{proof}
\begin{remark} This  result appears in \cite{Murphy}; the proof here  does
not need theorem  \ref{AH} or properties of the mirror map.
\end{remark}

\begin{proof} [Proof of theorem \ref{braidsum}] Write
\[\Pi_m=\sum_{{i=0}\atop{i+j=m}}^{m-1}A_{i,j-1}.\] Then
\[\sum_{{i=1}\atop{i+j=m}}^{m-1}i(A_{i,j-1}-A_{i-1,j})+\Pi_m=mA_m,\] giving
\[\Pi_m=mA_m-z\sum_{{i=1}\atop{i+j=m}}^{m-1}iA_i\overline{A}_j.\] This is the
coefficient of $t^m$ in
$\left(\sum^\infty_{i=1}iA_it^i\right)\overline{A}(t)$. Now \[z
\sum^\infty_{i=1}iA_it^i=t\frac{d}{dt}(A(t)).\] We then have
\begin{eqnarray}
z\sum_{m=1}^\infty\Pi_mt^{m-1}&=&\frac{d}{dt}(A(t))\overline{A}(t)\nonumber\\
&=&\frac{d}{dt}\ln(A(t)),\label{log}
\end{eqnarray} since $\overline{A}(t)=A(t)^{-1}$ by corollary \ref{mirror}.
Now \begin{eqnarray} \ln A(t)&=& \ln H(st)-\ln H(s^{-1}t),\nonumber\\ &=&
\sum_{m=1}^\infty (s^m-s^{-m})\frac{P_m}{m}t^m,\label{Pm}
\end{eqnarray} by theorem \ref{AH} and the definition of $P_m$. Comparing the
coefficients of $t^{m-1}$ in (\ref{log}), using(\ref{Pm}),  gives
\[(s-s^{-1})\Pi_m=(s^m-s^{-m})P_m,\] and hence the result.
\end{proof}








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\bibitem{Dipper}  {\bf R\,Dipper}, {\bf  G\,D\,James}, {\it Blocks and
idempotents of Hecke algebras of general linear groups}, 
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\bibitem{Hadji} {\bf R\,J\,Hadji}, {\it Knots, tangles and algebras}. MSc dissertation, University of Liverpool (1999)

\bibitem{Kosuda} {\bf M\,Kosuda}, {\bf J\,Murakami}, {\it Centralizer algebras of the mixed tensor representations of quantum group $U\sb q({\rm gl}(n,C))$}, 
{Osaka J. Math.} 30 (1993) 475--507

\bibitem{Macdonald} {\bf I\,G\,Macdonald}, {\it Symmetric functions and {H}all
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\bibitem{AistonMorton} {\bf H\,R\,Morton}, {\bf A\,K\,Aiston}, 
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\bibitem{Murphy} {\bf H\,R\,Morton},  {\it Skein theory and the {M}urphy operators}, 
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