}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-42.59435pt}{-18.84721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}{ {}{}{}}{{{{}}{{}}{{}}{{}}}{{{{}}{
{}{}}{}{}{{}{}}}} }{{{{}}{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}}
}{{}{}}{{}} {}{}{}{{{}}{{}}{{}}} {{{}}{{}}{{}}}
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}{}{{}}{}{}{}{}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-35.19012pt}{-16.43196pt}\pgfsys@curveto{-28.12857pt}{-2.40866pt}{-20.31795pt}{4.94434pt}{-6.26184pt}{10.98961pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.91864}{0.3951}{-0.3951}{0.91864}{-6.07814pt}{11.06862pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-42.78381pt}{4.3931pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\mathscr{N}(\Sigma_{i})}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{ {}{}{}}{}{ {}{}{}}{
{}{}{}}{{{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}}
}{{{{}}{{}}{{}}{{}}{{}}}{{{{}}{ {}{}}{}{}{{}{}}}} }{{}{}}{{}}
{}{}{}{{{}}{{}}{{}}} {{{}}{{}}{{}}}
{}{{}}{}{{}}{}{{}}{}{}{}{}{}{}{}{{}}{}{{}}{}{}{}{}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-30.60551pt}{-23.78923pt}\pgfsys@curveto{-12.31985pt}{-39.13297pt}{12.31985pt}{-39.13297pt}{30.29913pt}{-24.04631pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.76604}{0.64279}{-0.64279}{0.76604}{30.45232pt}{-23.91777pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-16.66972pt}{-42.8998pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\mathscr{N}(\Sigma_{i}\cup\Sigma_{i+1})}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{42.39436pt}{-18.84721pt}\pgfsys@lineto{65.59438pt}{-18.84721pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{65.79436pt}{-18.84721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{82.50542pt}{-18.84721pt}\pgfsys@lineto{105.70543pt}{-18.84721pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{105.90541pt}{-18.84721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{85.32623pt}{-14.74445pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\mathscr{N}(\Sigma_{k})}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

where $L$ is the circle removal correspondence defined above (and
corresponding to the identification 2-morphism).

#### General 2-handle attachmnents

Assume now that $Y\colon\underline{\Sigma}\Rightarrow\underline{S}$, is a
compression body, and let

$\underline{M}=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.33124pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{G_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.55417pt\raise 6.0361pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.74722pt\hbox{$\scriptstyle{M_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.33124pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.33124pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
52.42603pt\raise 6.0361pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.74722pt\hbox{$\scriptstyle{M_{2}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 72.99371pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
72.99371pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
92.17046pt\raise 6.0861pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.69722pt\hbox{$\scriptstyle{M_{k}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 110.49371pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
110.49371pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{G_{k}.}$}}}}}}}\ignorespaces}}}}\ignorespaces$
$\underline{N}=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.33124pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{G_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.1375pt\raise 6.0361pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.74722pt\hbox{$\scriptstyle{N_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.33124pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.33124pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
53.00935pt\raise 6.0361pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.74722pt\hbox{$\scriptstyle{N_{2}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 72.99371pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
72.99371pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
92.7538pt\raise 6.0861pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.69722pt\hbox{$\scriptstyle{N_{k}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 110.49371pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
110.49371pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{G_{k}.}$}}}}}}}\ignorespaces}}}}\ignorespaces$

be the sequences corresponding respectively to $\underline{\Sigma}$ and
$\underline{S}$. Associate then to $Y$ the following diagram $\mathcal{D}(Y)$:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.33124pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{G_{0}\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}{}{{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
12.55417pt\raise 13.75pt\hbox{\hbox{\kern
3.0pt\raise-1.74722pt\hbox{$\textstyle{\scriptstyle
M_{1}}$}}}}}\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
32.33347pt\raise 5.70071pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}{}{{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
13.1375pt\raise-13.75pt\hbox{\hbox{\kern
3.0pt\raise-1.74722pt\hbox{$\textstyle{\scriptstyle
N_{1}}$}}}}}\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
33.98141pt\raise-4.80586pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
20.33124pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\hbox{{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise
5.0pt\hbox{\hbox{\line@@}}}}}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise
5.0pt\hbox{\hbox{\line@@}}}}}}}}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise
2.5pt\hbox{\hbox{\line@@}}}}}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise
2.5pt\hbox{\hbox{\line@@}}}}}}}}{\hbox{\kern
0.0pt\hbox{\ignorespaces\hbox{\kern
0.0pt\raise-5.0pt\hbox{\lx@xy@tip{1.5}\lx<EMAIL_ADDRESS>19.94926pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise-1.74722pt\hbox{$\textstyle{\scriptstyle
L_{1}}$}}}}}\ignorespaces{\hbox{\kern 32.33124pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{G_{1}\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}{}{{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
53.21664pt\raise 13.75pt\hbox{\hbox{\kern
3.0pt\raise-1.74722pt\hbox{$\textstyle{\scriptstyle
M_{2}}$}}}}}\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
72.99594pt\raise 5.70071pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}{}{{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
53.79997pt\raise-13.75pt\hbox{\hbox{\kern
3.0pt\raise-1.74722pt\hbox{$\textstyle{\scriptstyle
N_{2}}$}}}}}\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
74.64389pt\raise-4.80586pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
60.99371pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\hbox{{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise
5.0pt\hbox{\hbox{\line@@}}}}}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise
5.0pt\hbox{\hbox{\line@@}}}}}}}}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise
2.5pt\hbox{\hbox{\line@@}}}}}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise
2.5pt\hbox{\hbox{\line@@}}}}}}}}{\hbox{\kern
0.0pt\hbox{\ignorespaces\hbox{\kern
0.0pt\raise-5.0pt\hbox{\lx@xy@tip{1.5}\lx<EMAIL_ADDRESS>60.61174pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise-1.74722pt\hbox{$\textstyle{\scriptstyle
L_{2}}$}}}}}\ignorespaces{\hbox{\kern 72.99371pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{G_{2}\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}{}{{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
113.66055pt\raise 4.9985pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}{}{{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
113.66055pt\raise-4.9985pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
100.86557pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\hbox{{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise
5.0pt\hbox{\hbox{\line@@}}}}}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise
5.0pt\hbox{\hbox{\line@@}}}}}}}}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\kern 0.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise
2.5pt\hbox{\hbox{\line@@}}}}}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise
2.5pt\hbox{\hbox{\line@@}}}}}}}}{\hbox{\kern
0.0pt\hbox{\ignorespaces\hbox{\kern
0.0pt\raise-5.0pt\hbox{\lx@xy@tip{1.5}\lx<EMAIL_ADDRESS>113.65619pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{{}{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}{}{{{}{}{}{{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}}}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
151.16055pt\raise 6.73419pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
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where $L_{i}=L(\widetilde{Y}_{i})$ stands for the correspondence of definition
5.11, with $Y_{i}\subset Y$ the piece flowing from $\Sigma_{i}$ to $S_{i}$,
and $\widetilde{Y}_{i}$ obtained by modding out the vertical tubes (i.e. the
tubes flowing between the circles of the decompositions $\underline{\Sigma}$
and $\underline{S}$) by the gradient flow of the Morse function, i.e.
collapsing them to circles.

######  Lemma 5.15.

(Independence of functions/pseudo-gradient) The diagram $\mathcal{D}(Y)$, as a
2-morphism of $\widehat{\mathcal{H}am}$, is independent on the choice of Morse
functions and pseudo-gradients.

###### Proof.

First, the choice of a different pseudo-gradient can have the effect of
twisting the vertical tubes, which might have the effect of changing the
correspondences $L_{i}$, but not the whole diagram, as can be shown by a
similar argument as in lemma 5.10.

Observe then that since they all are of the same index, the number of critical
points on each piece $Y_{i}$ is determined by the decomposition of $\Sigma$
and $S$.

Notice finally that the Lagrangian correspondence only depend on the attaching
circles. And they change just by isotopies or handleslides, which has no
effect on $L_{i}$.

∎

###  5.4. Cerf moves invariance

We now prove that two decompositions of a given cobordism yield equivalent
morphisms of $\widehat{\mathcal{H}am}$, in the sense of definition 5.8. Before
starting, we point out that the weak transversality assumption will always be
satisfied in our case. Indeed, $L$ and $(\prod{\underline{P}})\times C$ will
always be irreducible real algebraic affine varieties, therefore the only way
that $L$ could fail to intersect $(\prod{\underline{P}})\times C$ weakly
transversely would be to be entirely contained in it, which is never the case,
since in the holonomy descriptions, $L$ is defined by equations such as
$\mathrm{Hol}_{\gamma}A=1$ for some curves $\gamma$, while $C$ is defined by
equations such as $\mathrm{Hol}_{\gamma}A=-1$: the trivial representation is
always contained in $L$, but not in $C$.

At the level of 1-morphisms, the circle insertion/removal invariance follows
from proposition 4.5 and the definition of the equivalence relation.

We now check the moves for 2-morphisms.

#### Diffeomorphism equivalence

Follows from the fact that a diffeomorphism of surfaces induces a
symplectomorphism on moduli spaces, that preserves the Hamiltonian actions,
and a diffeomorphism of 3-manifolds maps Lagrangian correspondences to
Lagrangian correspondences.

#### Cylinder creation/cancellation

Follows from the fact that the diagram associated to a cylinder is an identity
in $\widehat{\mathcal{H}am}$.

#### Circle insertion/removal

Follows from the definition of the equivalence relation in
$\widehat{\mathcal{H}am}$ (definition 5.8), and the fact that the diagram
associated to a circle removal is an identification 2-morphism, modulo this
relation.

#### Imbrication of compression bodies

Follows immediately from proposition 5.13.

#### Critical point switches

There are several cases to be distinguished, depending on the index of the
critical points.

When both index are equal to either 1 or 2, this is a special case of
imbrication of compression bodies.

When one of the critical points has index either 0 or 3, its attaching sphere
(or attaching belt) is a sphere (with some circles in it) and in particular a
whole connected component of the total space of $\underline{\Sigma}$. It
follows that the diagrams associated to the handle attachments correspond to
attaching 2-cells to the sequence $\underline{M}$ associated to
$\underline{S}$ in a way that does not overlap. Therefore the order of
attachment doesn’t matter.

It remains to check the case when the indexes are 1 and 2.

Let $S_{1},S_{2}\subset\underline{\Sigma}$ denote respectively a 0-shpere and
a 1-sphere, disjoint from each other. If $S_{1}$ and $S_{2}$ lie in different
components of the sequence $\underline{\Sigma}$, invariance follows from the
same reasons as in the previous case. If this is not the case, one can insert
circles separating them (by circle insertion invariance), so that they lie on
distinct components.

#### Index 0-1 (and 2-3) handle cancellation

Let $(Y_{i},Y_{i+1},Y_{i+2})$ be as in proposition 5.5. First,
$\mathscr{N}(D^{0})\simeq\mathscr{N}(D^{1})\simeq pt$, and the diagram
associated to $Y_{i}$ consists in inserting the correspondence
$pt\subset\mathscr{N}(D^{0})\times\mathscr{N}(D^{1})$. Observe then that
$\mathscr{N}(\Sigma_{i}^{1})$ and
$\mathscr{N}(D^{0}\cup(D^{1}\sharp\Sigma_{i}^{1}))$ are identified, and both
correspond to the symplectic quotient of
$\mathscr{N}(D^{1}\sharp\Sigma_{i}^{1})$, by the $SU(2)$-action on $\partial
D^{1}$. The diagrams associated to $Y_{i+1}$ and $Y_{i+2}$ both correspond
(modulo the equivalence relation of definition 5.8) to inserting the
identification 2-morphism for this reduction. It follows that the diagram
associated to $(Y_{i},Y_{i+1},Y_{i+2})$ is an identity, modulo the equivalence
relation.

Reversing the diagram, we obtain the index 2-3 cancellation move.

#### Index 1-2 handle cancellation

Assume that $Y_{i}$ and $Y_{i+1}$ correspond to a handle cancellation pair,
i.e. (the opposite of) $Y_{i}$ and $Y_{i+1}$ correspond to 2-handle
attachments along two closed curves that intersect transversely at a single
point. We can assume that these curves correspond respectively to $\alpha_{1}$
and $\beta_{1}$, with $\alpha_{1}$, $\beta_{1}$, …, $\alpha_{g}$, $\beta_{g}$
a symplectic basis of the fundamental group of the intermediate surface
$\Sigma$ (with the $\gamma$ curves removed).

These two curves define two coisotropic submanifolds of $\mathscr{N}(\Sigma)$

$\displaystyle C_{1}$ $\displaystyle=\\{A_{1}=1\\}\text{, and}$ $\displaystyle
C_{2}$ $\displaystyle=\\{B_{1}=1\\}$

that induce respectively $L(Y_{1})^{T}$ and $L(Y_{2})$. Since they intersect
transversely, it follows that their composition is embedded, and corresponds
to $L(Y_{i}\cup Y_{i+1})$, i.e. the diagonal correspondence.

##  6\. Future directions

We now outline some directions we plan to take in the future.

###  6.1. Existence of a (non-quasi) 2-functor, quasi-Hamiltonian analogue

By using the spaces $\mathscr{N}(\Sigma)$, we were able to construct a quasi
2-functor. A natural question follows:

######  Question 6.1.

Is it possible to replace the spaces $\mathscr{N}(\Sigma)$ in our construction
by suitable Hamiltonian manifolds, satisfying a gluing equals reduction
principle in a strict sense (i.e. without $C$ in proposition 4.5), so to
obtain a 2-functor $\mathcal{C}ob_{1+1+1}\to\widehat{\mathcal{H}am}$ that
still assigns to a closed surface a sequence whose composition is the
$SU(2)$-character variety?

We believe this question can be answered partially positively, at least in two
ways. First, by using the quasi-Hamiltonian spaces constructed in [AMM98].
These spaces satisfy a gluing equals reduction principle, but their moment map
takes values in the group, rather than its Lie algebra. This would lead to the
definition of an analogous partial 2-category $q\mathcal{H}am$. However, it
seems not obvious to define Floer homology in this setting, since these are
not symplectic manifolds.

Another possible solution would be to use the infinite dimensional moduli
spaces $\mathscr{M}(\Sigma)$ introduced by Donaldson [Don92]: these are moduli
spaces of flat connexions on $\Sigma$, but the restriction to the boundary may
not be constant, and defines a map
$A_{|\partial\Sigma}\colon\partial\Sigma\to\mathfrak{g}$. This moduli space is
acted on by the gauge group of $\partial\Sigma$, which identifies with $k$
copies of the loop group $LSU(2)$.

###  6.2. Extension to dimension zero

Since the former spaces $\mathscr{M}(\Sigma)$ are infinite dimensional, Floer
homology seems also difficult to define in this setting. However, it seems
possible to use these spaces in order to define a theory extended to dimension
zero: to a closed interval $I$ one can associate the path group
$\mathscr{G}(I)=Map(I,SU(2))$, which comes with an evaluation map to the
boundary of $I$: $ev\colon\mathscr{G}(I)\to SU(2)^{2}$. One can use these
evaluation maps to glue the groups: if $S^{1}=I\cup J$ is a decomposition of
the circle into two intervals, one has

$\mathscr{G}(S^{1})=\mathscr{G}(I)\times_{ev}\mathscr{G}(J).$

This suggests the definition of a partial 3-category, where objects would be
finite dimensional Lie groups, 1-morphisms Banach Lie groups together with
morphisms similar with these evaluation maps, 2-morphisms Banach Hamiltonian
manifolds, and 3-morphisms $\mathscr{G}$-Lagrangian correspondences.

###  6.3. Extension to dimension four

The aim of this project is to promote $\widehat{\mathcal{H}am}$, or at least
the pre-completion $\underline{\mathcal{H}am}$, to a 3-category, and extend
the quasi 2-functor defined here to a quasi 3-functor from
$\mathcal{C}ob_{1+1+1+1}$ (or a version enriched with cohomology classes).

If $G$ and $G^{\prime}$ are objects of $\mathcal{H}am$,
$\underline{M},\underline{M^{\prime}}\in\underline{hom}^{1}(G,G^{\prime})$,
and
$\mathcal{D},\mathcal{D}^{\prime}\in\underline{hom}^{2}(\underline{M},\underline{M^{\prime}})$,
one would like to define a 3-morphism space
$hom^{3}(\mathcal{D},\mathcal{D}^{\prime})$ using equivariant Floer homology
(or the chain complex defining it). Indeed, in such a situation, let

$\displaystyle\mathscr{M}$ $\displaystyle=\prod_{M}{M^{-}\times M},$
$\displaystyle\mathscr{L}_{0}$ $\displaystyle=\prod_{M}{\Delta_{M}},$
$\displaystyle\mathscr{L}_{1}$ $\displaystyle=\prod_{L}{L},$
$\displaystyle\mathscr{G}$ $\displaystyle=\prod_{G}{G},$

where $M$, $L$ and $G$ run respectively in the set of symplectic manifolds,
Lagrangian multi-correspondences and Lie groups appearing in the diagram
$\mathcal{D}\sharp_{\underline{M}\sharp\underline{M^{\prime}}}\mathcal{D}^{\prime}$.
One would then take

$hom^{3}(\mathcal{D},\mathcal{D}^{\prime})=CF_{\mathscr{G}}(\mathscr{L}_{0},\mathscr{L}_{1}).$

Several constructions of such chain complexes appeared in the literature, for
example [Fra04, HLS16], or a construction outlined in [DF17]. We plan to
define another version relying on Wehrheim and Woodward’s quilt theory, that
should be well suited for our purposes. The resulting algebraic structure
should be a 3-category analogue of Bottman and Carmeli’s
$(A_{\infty},2)$-categories [BC18].

###  6.4. Invariants for knots and sutured manifolds

The framework developped here should be well-suited for defining invariants of
knots, and more generally sutured manifolds, similar with the ones in
Heegaard-Floer theory, since a sutured manifold can be viewed as a 2-morphism
in $\mathcal{C}ob_{1+1+1}$. After applying the functor, one gets a 2-morphism
in $\widehat{\mathcal{H}am}$, which can be horizontally composed with the
coadjoint orbit $\left\\{\theta\in\mathfrak{g}\ |\
\left|\theta\right|=\frac{\pi}{2}\right\\}$. That precisely corresponds to
putting a traceless condition on the holonomy of a connexion around a meridian
of a knot, as in for example [KM11].

###  6.5. Relation with Seiberg-Witten theory

The ideas in this section emerged during a conversation with Guangbo Xu.
Denote

$F_{Don}\colon\mathcal{C}ob_{1+1+1}\dashrightarrow\widehat{\mathcal{H}am}$

the quasi 2-functor defined in this paper. One should be able to define a
similar quasi 2-functor

$F_{SW}\colon\mathcal{C}ob_{1+1+1}\dashrightarrow\widehat{\mathcal{H}am}$

that would correspond to Seiberg-Witten theory, using extended moduli spaces
of vortices analogous to the spaces $\mathscr{N}(\Sigma)$.

Following Witten’s conjecture, these two theories should be related: one can
expect that there is a natural transformation

$T\colon F_{Don}\to F_{SW},$

i.e. to any $k$-morphism $W\colon X\to Y$ in $\mathcal{C}ob_{1+1+1}$, should
correspond a $(k+1)$-morphism in $\widehat{\mathcal{H}am}$ relating
$F_{Don}(W)$ and $F_{SW}(W)$ as in the following diagram:

$\textstyle{F_{Don}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T(X)}$$\scriptstyle{F_{Don}(W)}$$\textstyle{F_{Don}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T(Y)}$$\textstyle{\Downarrow
T(W)}$$\textstyle{F_{SW}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{SW}(W)}$$\textstyle{F_{SW}(Y).}$

One can think of $T(W)$ as being associated with the $(k+1)$-morphism
$W\times[0,\infty]$, where the $[0,\infty]$ coordinate could correspond to a
wavelength. We plan to build $F_{SW}$ and $T$ in a future joint work, building
on the work of Feehan and Leness [FL01a, FL01b].

## References

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  * [AMM98] Anton Alekseev, Anton Malkin, and Eckhard Meinrenken.  Lie group valued moment maps.  J. Differential Geom., 48(3):445–495, 1998.
  * [BC18] Nathaniel Bottman and Shachar Carmeli.  $(A_{\infty},2)$-categories and relative 2-operads.  arXiv e-prints, page arXiv:1811.05442, November 2018.
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  * [CM18] Laurent Côté and Ciprian Manolescu.  A sheaf-theoretic SL(2,C) Floer homology for knots.  arXiv e-prints, page arXiv:1811.07000, Nov 2018.
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  * [Don92] S. K. Donaldson.  Boundary value problems for Yang-Mills fields.  J. Geom. Phys., 8(1-4):89–122, 1992.
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  * [FL01b] Paul M. N. Feehan and Thomas G. Leness.  $\rm PU(2)$ monopoles. II. Top-level Seiberg-Witten moduli spaces and Witten’s conjecture in low degrees.  J. Reine Angew. Math., 538:135–212, 2001.
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# Rapid Evolution of the White Dwarf Pulsar AR Scorpii

Peter Garnavich Department of Physics and Astronomy, University of Notre Dame,
Notre Dame, IN 46556, USA  Stephen B. Potter  South African Astronomical
Observatory, PO Box 9, Observatory 7935, Cape Town, South Africa  Department
of Physics, University of Johannesburg, PO Box 524, 2006 Auckland Park,
Johannesburg, South Africa  David A. H. Buckley  South African Astronomical
Observatory, PO Box 9, Observatory 7935, Cape Town, South Africa  Department
of Astronomy, University of Cape Town, Private Bag X3, Rondebosch 7701, South
Africa  Department of Physics, University of the Free State, PO Box 339,
Bloemfontein 9300, South Africa  Anke van Dyk South African Astronomical
Observatory, PO Box 9, Observatory 7935, Cape Town, South Africa  Department
of Astronomy, University of Cape Town, Private Bag X3, Rondebosch 7701, South
Africa  Daniel Egbo  South African Astronomical Observatory, PO Box 9,
Observatory 7935, Cape Town, South Africa  Department of Astronomy, University
of Cape Town, Private Bag X3, Rondebosch 7701, South Africa  Colin Littlefield
Bay Area Environmental Research Institute, Moffett Field, CA 94035 USA
Department of Physics and Astronomy, University of Notre Dame, Notre Dame, IN
46556, USA  Anousha Greiveldinger Department of Physics and Astronomy,
University of Notre Dame, Notre Dame, IN 46556, USA

###### Abstract

Analysis of AR Sco optical light curves spanning nine years show a secular
change in the relative amplitudes of the beat pulse pairs generated by the two
magnetic poles of its rotating white dwarf. Recent photometry now shows that
the primary and secondary beat pulses have similar amplitudes, while in 2015
the primary pulse was approximately twice that of the secondary peak. The
equalization in the beat pulse amplitudes is also seen in the linearly
polarized flux. This rapid evolution is consistent with precession of the
white dwarf spin axis. The observations imply that the pulse amplitudes cycle
over a period of $\gtrsim 40$ yrs, but that the upper limit is currently
poorly constrained. If precession is the mechanism driving the evolution, then
over the next 10 years the ratio of the beat pulse amplitudes will reach a
maximum followed by a return to asymmetric beat pulses.

cataclysmic variable stars; intermediate polars; white dwarf stars; magnetic
poles; rotation powered pulsars

##  1 Introduction

AR Scorpii (AR Sco hereafter) is one of the most intriguing interacting binary
stars known. It has been called a white dwarf “pulsar” (Buckley et al., 2017)
because its bright, polarized, synchrotron flashes appear to be powered by the
spin-down energy of its degenerate primary stellar component (Marsh et al.,
2016; Stiller et al., 2018; Gaibor et al., 2020; Pelisoli et al., 2022). The
binary consists of a low-mass red dwarf star and a rapidly spinning magnetized
white dwarf (WD) orbiting over a 3.56-hour period. The WD spins ($\omega$)
with a period of $1.95$ min (Marsh et al., 2016) and appears to generate two
pulses per rotation, likely from two magnetic poles. In the Potter & Buckley
(2018) model, the synchrotron emission is modulated by the angle between the
magnetic poles of the spinning WD and the secondary star. The emission is
enhanced just after a magnetic pole sweeps past the red dwarf, leading to
variations seen at the spin frequency plus strong pulses at the beat
frequency, $\omega-\Omega$, where $\Omega$ is the orbital frequency. Because
pulses are generated from both poles, power is also seen at the first harmonic
of the beat frequency (aka the double-beat: 2($\omega-\Omega$)).

Light curves of AR Sco obtained near its discovery clearly showed that the
beat pulse pairs alternated in strength with the brighter pole being about
twice the amplitude of the opposite pole (Marsh et al., 2016). This
alternating strength was seen in the linearly polarized flux as well (Potter &
Buckley, 2018). However, in a Fourier component analysis, Pelisoli et al.
(2022) noted that the quality of their model fit to the observed light curves
was decreasing over time. They attributed this to stochastic changes in the
relative strength of the pulse pairs. Takata et al. (2021) noted that the
X-ray beat pulses were single peaked in 2016 but seen as double-peaked in
2020. Here, we analyze nine years of AR Sco optical light curves to test for
any systematic evolution in the beat pulse strengths.

Table 1: Time-Series Photometry

Total Flux
---
Year | Start | Cadence | Length | Telescope | $R_{beat}^{a}$
| (MJD) | (sec) | (hours) |  |
2015 | 57197.9 | 1.3 | 2.8 | WHT | 0.42$\pm 0.03$
2016 | 57463.0 | 1.0 | 2.3 | SAAO | 0.53$\pm 0.03$
2016 | 57536.4 | 1.0 | 6.8 | SAAO | 0.49$\pm 0.03$
2017 | 57891.3 | 2.0 | 4.0 | SAAO | 0.37$\pm 0.05$
2018 | 58200.5 | 1.0 | 4.0 | SAAO | 0.58$\pm 0.03$
2019 | 58613.4 | 1.0 | 5.0 | SAAO | 0.66$\pm 0.03$
2020 | 59016.1 | 3.3 | 4.6 | SLKT | 0.69$\pm 0.04$
2020 | 59037.1 | 2.5 | 4.5 | SLKT | 0.62$\pm 0.03$
2021 | 59322.3 | 3.0 | 4.6 | SLKT | 0.65$\pm 0.03$
2021 | 59342.2 | 2.6 | 4.6 | SLKT | 0.77$\pm 0.04$
2022 | 59751.1 | 3.6 | 4.1 | SLKT | 0.79$\pm 0.03$
2023 | 60090.2 | 3.9 | 4.2 | SLKT | 0.79$\pm 0.04$
2023 | 60091.3 | 2.0 | 4.5 | SAAO | 0.85$\pm 0.03$
2023 | 60092.4 | 2.4 | 4.9 | SAAO | 0.82$\pm 0.03$
Linearly Polarized Flux
2016 | 57463.0 | 1.0 | 2.3 | SAAO | 0.52$\pm 0.03$
2023 | 60091.3 | 2.0 | 4.5 | SAAO | 0.81$\pm 0.03$

aafootnotetext: Beat/Double-Beat amplitude ratio calculated from Equation 1.

##  2 Data

We analyze light curves studied by Gaibor et al. (2020). In addition, we have
obtained rapid-cadence light curves of AR Sco using the Sarah L. Krizmanich
Telescope (SLKT) from 2020 through 2023 and a light curves in 2023 from the
South African Astronomical Observatory using the high-speed photomultiplier
instrument (HIPPO, Potter et al. (2008)). We analyze both the total flux and
polarized flux for two epochs obtained in 2016 and 2023 using the HIPPO
instrument. Only light curves covering a substantial fraction of a binary
orbit are included in this study. Properties of the photometric time-series
datasets are listed in Table 1.

For each photometric time-series, we constructed Lomb-Scargle periodograms
(L-S hereafter, Lomb, 1976; Scargle, 1982) implemented using the _LombScargle_
package in _astropy_ (Astropy Collaboration et al., 2018). For example, L-S
periodograms of the ULTRAcam photometry taken in 2015 and the SAAO photometry
from 2023 are displayed in Figure 1. From the strongest peaks in the
periodogram ($\omega-\Omega$, $2(\omega-\Omega)$, $2\omega-\Omega$,
$3(\omega-\Omega)$, $4(\omega-\Omega)$, $2\omega$, $2(2\omega-\Omega)$), we
reconstructed the 2015 and 2023 light curves and these are also shown in
Figure 1. The properties of the optical light curves appear to have changed
significantly between 2015 and 2023. In particular, the strength of the
‘secondary’ beat pulse has increased relative to the main pulse over this
period.

To quantify the relative strengths of the beat pulses averaged over an orbit,
we use the L-S periodogram peaks at the beat frequency ($A_{(\omega-\Omega)}$)
and the double-beat frequency ($A^{\prime}_{2(\omega-\Omega)}$), to define the
ratio:

$R_{beat}={{A^{\prime}_{2(\omega-\Omega)}}\over{A_{(\omega-\Omega)}+A^{\prime}_{2(\omega-\Omega)}}}\;\;.$
(1)

When $R_{beat}\approx 0$, only a single beat pulse is detected over a beat
period. Alternatively, when $R_{beat}$ approaches unity, the two beat pulses
are similar in amplitude. Here,
$A^{\prime}_{2(\omega-\Omega)}=A_{2(\omega-\Omega})-H_{2}\,A_{(\omega-\Omega)}$
is the amplitude of the double beat peak corrected for a contribution of the
second harmonic of the beat frequency. The parameter $H_{2}$ is the ratio
between the amplitudes of the beat frequency, and its second harmonic.
Simulated light curves show that an $H_{2}=0.30$ accounts for this harmonic
contribution111Fourier analysis of Gaussian pulses in flux give a closed form
for the harmonic ratio of $H_{2}=exp(-6\pi^{2}\sigma^{2}/P^{2})$, where $P$ is
the time between pulses. However, analyzing light curves in magnitudes
requires simulations to estimate $H_{2}$. that only becomes significant when
when $R_{beat}<0.5$, that is, when the secondary pulse is weak.

Figure 1: Left Column: The L-S periodogram (top) and section of the light
curve (bottom) from 2015. The periodogram displays nearly equal amplitude
peaks of the beat and double beat. This is seen in the light curve as the
primary peak being twice the amplitude of the secondary one. The Fourier model
is plotted as a solid black line and the data in faint red. Right Column: The
L-S periodogram (top) and light curve (bottom) from 2023. The periodogram
shows a very weak beat amplitude and a strong double beat peak. This is seen
in the light curve as nearly equal amplitude beat pairs.

The measured beat pulse ratios, $R_{beat}$, for 14 light curves are given in
Table 1. Uncertainties on the ratio measurements were estimated by taking L-S
periodograms of subsets of each light curve. The variance in $R_{beat}$ was
then used to calculate the errorbars shown. We also estimated the beat pulse
ratio in linearly polarized flux using photometric time-series obtained in
2016 and 2023 and these results are shown in Table 1.

##  3 Discussion

Figure 2 displays the AR Sco beat pulse ratio over time. While there is
scatter in the ratio from night to night, the trend is for an increasing ratio
consistent with the pulse pairs evolving to nearly equal strength. The
$R_{beat}$ parameter for the linearly polarized flux was also seen to increase
significantly between 2016 and 2023.

Interpreting the changes in pulse amplitude is difficult given that the source
of the relativistic electrons has not been established. Electrons may be
accelerated by direct interaction between the WD magnetic field and the field
of the secondary star (e.g. Takata et al., 2017; Garnavich et al., 2019). Slow
changes in the secondary star’s magnetic field could influence the WD field
lines involved in trapping the emitting electrons.

Figure 2: The ratio of the double-beat to beat amplitudes measured for 14
light curves (solid circles). The $R_{beat}$ parameter measured from the
linearly polarized flux in two epochs is displayed as open diamonds. The lines
show a range cyclical models with periods between 30 and 100 yr. The
$\chi^{2}$ parameter increases sharply for models with periods shorter than 40
years, but long periods are poorly constrained without further observations.

Katz (2017) proposed that the spin axis of the WD could be tilted relative to
the orbital angular momentum vector, leading to a precession of the WD and its
magnetic field configuration. Katz predicted that WD precession might impact
the phase of the brightest point in the orbital modulation seen in AR Sco.
Peterson et al. (2019) did not detect a shift in the orbital modulation curve
using archival photometry222Also see Littlefield et al. (2017). However, the
narrowness of the synchroton beams could provide a sensitive test of
precessional motion if it results in relative changes in the viewing angles of
the two poles. Katz (2017) predicted precession periods between 20 and 200
years primarily depending on the WD mass.

Figure 3: The geometry of a precessing WD with synchrotron beams emitted from
magnetic poles. The magnetic axis makes an angle $\chi$ relative to the spin
axis (solid red lines). The binary orbital inclination is $i$, and the
obliquity of the spin axis is $\epsilon$. The path of spin axis precession is
indicated by dashed red circles. The angle between the direction to Earth and
the peak of a synchrotron beam is $\theta$. For simplicity, the synchrotron
beams are shown emanating from the WD, but they actually originate from the
magnetic fields on the opposite side of the WD (see Potter & Buckley, 2018).

###  3.1 Precession Model

To test if WD precession could generate the observed variation in pulse
amplitudes, we constructed a simple model illustrated in Figure 3. The model
parameters include the binary inclination, $i$, the obliquity of the spin
axis, $\epsilon$, and the angle between the magnetic dipole and the WD spin
axis, $\chi$. From this geometric model, we calculate the minimum angle,
$\theta$, between center of the synchrotron beams and the direction to the
Earth and follow its variation over a precession cycle.

Based on the estimated binary mass ratio (Marsh et al., 2016), and lack of
detected eclipses the orbital inclination is $i\approx 75^{\circ}$ (Garnavich
et al., 2019). That there are two pulses per WD spin suggests that the
magnetic field axis is highly inclined to the spin axis (Geng et al., 2016),
however, the symmetry of truly perpendicular rotator ($\chi>80^{\circ}$) would
severely limit the range of pulse ratio variations over a precession cycle.

Following the beam model described in Potter & Buckley (2018), we approximate
the synchrotron pulse profile by a Gaussian function with $\sigma=40^{\circ}$.
Thus, the intensity of the synchrotron beam will have fallen to half its peak
value when viewed at an angle of $\theta\approx 47^{\circ}$. For simplicity,
we assume that the peak intensity and profile distributions are identical for
the two emitting poles.

We find that the observed change in $R_{beat}$ from 0.5 to 0.8 can be achieved
by the precession model using reasonable values of the parameters (Figure 4).
A WD obliquity near $\epsilon\approx 10^{\circ}$ is sufficient to reproduce
the observations. Interestingly, when the obliquity exceeds the co-inclination
($90-i$), the secondary pulse strength can exceed the amplitude of the primary
pulse, as shown in the lower panel of Figure 4.

Besides the $R_{beat}$ ratio parameter, an additional observational constraint
on precession models is a direct measurement of the amplitude changes of the
pulses from each pole. As displayed in Figure 4, the amplitude of the pulse
from the dominant pole varies by only 10% over a quarter of the precession
cycle for the first set of model parameters. In general, this small amplitude
change for one pole results when the obliquity is low and $\chi\approx i$. For
this geometry, changes in $\theta$ for one of the emitting poles remains small
over a precession cycle, while the second pole creates most of the variation
in the amplitude ratio.

Figure 4: Predictions for the $R_{beat}$ parameter as a function of precession
phase for two sets of model parameters. Top: The relative pulse amplitudes
over two precession cycles for a WD obliquity of 10∘. The right panel shows
that the resulting $R_{beat}$ parameter varies nearly linearly between 0.5 and
0.8 over half of a precession period. Bottom: The same model except the
obliquity has been increased to 20∘ and now exceeds the co-inclination of the
orbit. In this case, the poles can switch in dominance and create sharp
features in the $R_{beat}$ curve.

The lower panels of Figure 1 show that the beat pulse amplitude of the
dominant pole remained nearly the same between 2015 and 2023, and that most of
the $R_{beat}$ evolution came from strengthening in the secondary pulse. We
estimate that the amplitude of the primary pulse varied by no more than 0.1
mag over the nine years of observation. We therefore infer that $\chi\approx
i$ for AR Sco.

###  3.2 Precession Period

Despite the rapid variation observed in $R_{beat}$ over nine years, the
apparent linear increase with time poorly constrains estimates of the
precession period. Figure 2 displays sinusoidal functions with periods between
30 and 100 yr that have been fitted by minimizing the $\chi^{2}$ parameter.
The $\chi^{2}$ parameter steeply increases for periods less than 40 yr, but
$\chi^{2}$ is nearly constant for models with $P>40$ yr. A linear fit of the
$R_{beat}$ ratio shows it rising by 0.05 per year, meaning that the pulses
could reach parity as early as the year 2027 (MJD$\approx$61400). Sinusoidal
extrapolations indicate that $R_{beat}$ will reach a maximum before the year
2029.

##  4 Conclusion

The AR Sco beat pulse pairs have evolved from a strong asymmetry to become
nearly equal in amplitude over a decade of observations. This evolution is
supported by the changes noted in the X-ray beat pulse noted by Takata et al.
(2021). We also find that the beat pulses have evolved in linearly polarized
flux and that their amplitudes are now nearly equal. This suggests that the
evolution results from physical changes or viewing angle variations in the
highly polarized synchrotron beams.

The precession model suggests that in 2023 we were viewing the WD spin axis
oriented so that the synchrotron beams are seen symmetrically, while in 2015,
precession of the spin axis resulted in a tilt that favored our view of one of
the magnetic poles. If precession is the origin of this evolution, then over
the next 10 years the ratio of the beat pulse amplitudes, $R_{beat}$, will
reach a maximum followed by a return to asymmetric beat pulses.

We dedicate this study to Tom Marsh. PMG thanks the Krizmanich family for
their generous donations for the construction and support of the Sarah L.
Krizmanich Telescope.

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# Cluster-level Group Representativity Fairness in $k$-means Clustering

Stanley Simoes School of EEECS, Queen’s University Belfast, UK
<EMAIL_ADDRESS>Deepak P School of EEECS, Queen’s University Belfast, UK
<EMAIL_ADDRESS>Muiris MacCarthaigh School of HAPP, Queen’s University
Belfast, UK<EMAIL_ADDRESS>
###### Abstract

There has been much interest recently in developing fair clustering algorithms
that seek to do justice to the representation of groups defined along
sensitive attributes such as race and gender. We observe that clustering
algorithms could generate clusters such that different groups are
disadvantaged within different clusters. We develop a clustering algorithm,
building upon the centroid clustering paradigm pioneered by classical
algorithms such as $k$-means, where we focus on mitigating the unfairness
experienced by the most-disadvantaged group within each cluster. Our method
uses an iterative optimisation paradigm whereby an initial cluster assignment
is modified by reassigning objects to clusters such that the worst-off
sensitive group within each cluster is benefitted. We demonstrate the
effectiveness of our method through extensive empirical evaluations over a
novel evaluation metric on real-world datasets. Specifically, we show that our
method is effective in enhancing cluster-level group representativity fairness
significantly at low impact on cluster coherence.

### Keywords

algorithmic fairness, unsupervised machine learning, clustering,
representativity

##  1 Introduction

Fairness in ML [13] has seen much scholarly activity in recent times. Within
the broader umbrella of fair ML, several fair clustering methods have also
been developed [11]. Most of these endeavours, starting from a pioneering work
by Chierichetti et al [12], have considered ensuring proportional
representation of sensitive groups – such as those defined on race and gender
– within each cluster in the output; this is often referred to as group
fairness [16]. Recent formulations of fair clustering are able to incorporate
considerations to fairness along multiple sensitive attributes together [2].
Such representational fairness of sensitive groups may be seen as the
application of the notion of proportional representation, aka statistical
parity [7], within clustering.

It may be argued that simply ensuring representation of sensitive groups
within each cluster is insufficient, especially within the popular paradigm of
centroid clustering, pioneered by the $k$-means clustering algorithm [24]. In
centroid clustering, each cluster is represented by a prototype, often
informally referred to as the centroid. A data object’s proximity to its
cluster centroid is a key criterion determining the quality of the clustering.
For example, an object that happens to be very close to its centroid could be
thought of as being ‘better represented’ within the clustering than another
that happens to be much further away from its centroid. Recent work on fair
clustering has sought to deepen uniformity of distance-to-centroid, dubbed as
representativity fairness [26], across all dataset objects. Representativity
fairness seeks to ensure deeper levels of uniformity of representativity
(_i.e._ , centroid distance) across objects, regardless of their sensitive
attribute membership, and may thus be regarded as an instantiation of
individual fairness [16]. The notion of representativity has been extended to
the framework of sensitive groups independently by Abbasi et al [1] and
Ghadiri et al [19]. While the methodologies they propose are different, they
are strikingly similar in the nature of the fairness sought. They consider the
aggregate representativity (they use the term representation) across objects
within each sensitive group, and target proportionality along such group-
specific aggregates across sensitive groups. For example, within a social
media profile clustering scenario, this implies that the mean distance of
female profiles from their centroids should be as close as possible to the
mean distance of male profiles from their centroids. This fairness condition
targets that groups are fairly treated across all the clusters in the output
cluster assignment.

Figure 1: Illustration of representativity across differently sized clusters

Against this backdrop, we note two issues with such representativity
aggregations across sensitive groups and clusters. First, note that
representativity is quantified as the distance of an object to its cluster
centroid111Representativity, in a literal sense, may be argued to mean
centroid proximity than centroid distance; however, since it is easier to deal
with distances, we consistently refer minimisation of centroid distance as
representativity enhancement, which is essentially maximising centroid
proximity.. This construction of representativity, we observe, does not yield
well to cross-cluster comparisons, especially when there could be clusters
with widely varying sizes. Consider the toy 2-cluster clustering in Figure 1
where the black and circled white objects represent different sensitive
groups, with the white object representing the centroid; also assume that the
grey area is full of other objects which we do not consider for now. The black
object has better representativity (_i.e._ , better centroid proximity) in the
larger cluster, and vice versa. However, given the size of the clusters, the
same numeric difference in representativity should be regarded as almost
insignificant in the larger cluster, and very consequential in the smaller
cluster, when conditioned on the relative cluster sizes. In other words,
considering representativity against the backdrop of the cluster sizes, we may
assume that the circled white group gets a much better overall
representativity than the black group. In sharp contrast to such intuitive
judgement, observe that the simple aggregate of representativity misleadingly
puts the black and circled white groups on an equal footing. Second, note that
the fairness goal seeks to bring about parity across sensitive groups. For a
clustering task, as in the case of earlier work on statistical parity for
group fairness (_e.g._ , Chierichetti et al [12], Abraham et al [2]), the
natural granularity for evaluating clustering fairness is at the level of each
cluster. The cross-cluster aggregation of representativities is neither
aligned with pragmatism (_e.g._ , decision scenarios almost always involve
cluster-level decision-making [25]) or underpinning political justifications
(_e.g._ , in contrast to extant justifications of statistical parity [20]). We
additionally note that the same sensitive group may be advantaged in one
cluster and disadvantaged in another. The paradigm of cross-cluster
aggregation of representativities allows such effects to cancel out, thus
acting as a veneer to conceal potentially deep levels of intra-cluster
unfairness.

###  1.1 Our Contributions

We propose a novel formulation of group fairness extending the notion of
representativity fairness along sensitive groupings in the data. Our cluster-
level group representativity fairness targets cluster-level fairness, thus
mitigating identified cross-cluster representativity aggregation issues. We
propose a novel fair clustering method targeting to optimise for cluster-level
group representativity fairness, and illustrate, through extensive empirical
evaluations, that our method is able to achieve high degrees of fairness on
appropriate evaluation metrics.

##  2 Related Work

We now briefly summarise research on group fairness and individual fairness in
fair clustering.

###  2.1 Group Fairness

The notion of group fairness, which targets proportional representation of
sensitive attribute groups, was pioneered by Chierichetti et al [12] in 2017.
Since then, group fair clustering research has diversified into considering
settings such as spectral clustering [22], hierarchical clustering [4] and
probabilistic scenarios [17]. Along another facet, variants of group fairness
notions such as capped representation [3] and equitable group-level
representativity [1, 19] have been explored. There have also been fairness
conceptualisations such as proportional fairness [10] that straddle boundaries
between group and individual fairness.

###  2.2 Individual Fairness

At a risk of overgeneralisation, one may paraphrase individual fairness as
ensuring that similar objects be given similar outcomes (_i.e._ , cluster
membership in clustering). While this is inarguably aligned with the
clustering objective of maximising intra-cluster similarities and minimising
inter-cluster similarities, clustering algorithms can only achieve the
objective on a best-effort basis, making some shortfall inevitable. Specific
formulations of individually fair clustering methods have explored curtailing
the shortfall through ways such as using randomness [8]. P & Abraham [26]
considers optimising for the uniformity of centroid-proximity (aka
representativity) across objects. The focus on representativity continues in
much recent work, with Vakilian & Yalciner [27] considering an object-specific
upper bound for representativity and Chakrabarti et al [9] proposing that
representativities between pairs of objects be bounded above by a specified
multiplication factor.

###  2.3 Positioning Our Work

Our notion of cluster-level group representativity fairness is a novel
conceptualisation of fairness unexplored in previous work. Our usage of
cluster-level group fairness quantification as an intermediate level between
individual and group makes it distinct from previous work on both fairness
streams.

##  3 Background

We briefly outline the formulation of the popular $k$-means clustering
algorithm [24], as essential background to describe our method. Consider a set
of relational data objects $X=\\{\ldots x\ldots\\}$ where
$x\in\mathbb{R}^{d}$, which need to be partitioned into $k$ clusters,
$\mathcal{C}=\\{\ldots C\ldots\\}$. $k$-means uses an EM-style [14]
optimisation framework to optimise for:

(3.1) $\sum_{C\in\mathcal{C}}\sum_{x\in C}||x-\mu_{C}||^{2}$

where $\mu_{C}\in\mathbb{R}^{d}$ is the centroid or prototype object of
cluster $C$. The reader would notice that the objective relates to a given
cluster assignment $\mathcal{C}$; the EM-style optimisation starts with a
given cluster assignment, and iteratively refines the cluster assignment and
centroids in order to minimise the objective in Equation 3.1.

As a generic clustering method, $k$-means is obviously agnostic to sensitive
attribute groups to which individual objects belong. Using the terminology of
representativity (_i.e._ , centroid proximity), one may observe that $k$-means
seeks to optimise for the sum of representativities across all objects in the
dataset. This may be regarded as a Benthamite [5] utilitarian objective that
seeks the greatest good for the greatest number. Abbasi et al [1] and Ghadiri
et al [19] note that this can result in cluster centroids representing
sensitive groups differently, often favouring one over the other. This would
have implications especially in cases where the centroids are used to
summarise objects in the clusters. To mitigate the disparity in
representativities of sensitive groups, Abbasi et al [1] and Ghadiri et al
[19] independently proposed a new fair $k$-means objective. Suppose each data
object in $X$ belongs to one of several groups $S$ (_e.g._ , female, male,
_etc._) defined across a sensitive attribute $\mathcal{S}=\\{\ldots
S\ldots\\}$ (_e.g._ , sex). The fair $k$-means objective [1, 19] seeks to
optimise for the worst-off sensitive group at the cluster assignment level:

(3.2) $\max_{S\in\mathcal{S}}\frac{1}{|S|}\sum_{C\in\mathcal{C}}\sum_{x\in
C\cap S}||x-\mu_{C}||^{2}$

The objective in Equation 3.2 suffers from a variant of the issue observed in
the $k$-means objective – the averaging property can allow a sensitive group
to have a low representativity in one cluster which is discounted by
moderately high representativities in the other clusters. Further, a single
sensitive group may not be the worst-off in all clusters. Towards this, we
turn our attention to mitigating group unfairness at the cluster level, rather
than the cluster assignment level.

##  4 Problem Statement

Our notion of cluster-level group representativity fairness targets to
minimise the disparity in the representativities of the sensitive groups
within each cluster. We assume a single sensitive attribute $\mathcal{S}$.
Consider a cluster $C$ within a cluster assignment $\mathcal{C}$. The
representativity loss of the set of objects taking $S$ within $C$ is denoted
as:

(4.3) $\overline{f}(S,C)=\frac{1}{|C\cap S|}\sum\limits_{x\in C\cap
S}||x-\mu_{C}||^{2}$

Thus for each cluster $C$, we obtain a _representativity vector_ with one
component per sensitive group. We say that a cluster assignment is fair if all
clusters in it are fair, and we say that a cluster is fair if its
corresponding representativity vector is uniform, _i.e._ , the gap between the
best-off and worst-off groups is as small as possible.

In view of this novel formulation of cluster-level group representativity
fairness, we consider developing a metric towards quantifying adherence to the
notion. In this paper, we look at the case where the number of sensitive
groups is not large. Accordingly, we capture the representativity disparity
between the best-off and worst-off groups simply by taking the difference in
their representativities, given by:

(4.4)
$\mathit{CD}(C)=\Bigg{(}\frac{\max\limits_{\begin{subarray}{c}S\in\mathcal{S}\end{subarray}}\overline{f}(S,C)-\min\limits_{\begin{subarray}{c}S\in\mathcal{S}\end{subarray}}\overline{f}(S,C)}{\min\limits_{\begin{subarray}{c}S\in\mathcal{S}\end{subarray}}\overline{f}(S,C)}\Bigg{)}\times
100$

$\mathit{CD}(C)$, or cluster disparity, quantifies the difference between the
representativity losses for the best-off and worst-off groups within cluster
$C$, expressed as a percentage. Evidently, we would ideally like
$\mathit{CD}(C)$ to be $0.0$ which is achieved when the same representativity
loss is experienced by each sensitive group. The formulation of
$\mathit{CD}(C)$ as a percentage allows a fair basis for aggregating this
across clusters of different sizes and shapes, thus addressing the first issue
with representativity aggregation pointed out in Section 1 – that of cluster
sizes. We aggregate this across clusters in the cluster assignment
$\mathcal{C}$ to arrive at a single measure for the overall cluster
assignment:

(4.5)
$\mathit{ACD}(\mathcal{C})=\frac{1}{|\mathcal{C}|}\sum_{C\in\mathcal{C}}\mathit{CD}(C)$

The average cluster disparity abbreviated as $\mathit{ACD}(\mathcal{C})$,
being an average of percentages, evaluates to a non-negative value, with a
lower value indicating a smaller disparity between the worst-off group and
best-off group in individual clusters, and consequently a fairer clustering.

##  5 Proposed Method

We now describe our fair clustering method focused on cluster-level group
representativity fairness. We start by describing our objective function
followed by the optimisation framework.

###  5.1 Objective Function

The notion of fairness outlined in Section 4 targets to enhance the uniformity
of the representativity vector for each cluster. This can be trivially
achieved by having a high representativity loss for all sensitive groups (as
noted by Abbasi et al [1]) which would make the resulting cluster assignment
of poor utility. Towards improving uniformity of the representativities, we
take cue from contemporary theories in political philosophy and focus on
mitigating the representativity loss (_i.e._ , $\overline{f}(.,.)$)
experienced by the worst-off group within each cluster. The worst-off group in
a cluster $C$ is given by:

(5.6) $\mathop{\arg\max}_{S\in\mathcal{S}}\ \ \overline{f}(S,C)$

This notion espouses the ethos across several popular philosophical theories
including the concern for the most vulnerable within the famed difference
principle [18] of distributive justice due to John
Rawls222https://en.wikipedia.org/wiki/John_Rawls.

Thus, the overall clustering objective we address here is to generate a
cluster assignment that comprises coherent clusters (the singular focus of
algorithms such as $k$-means) where, additionally, the representativity loss
for the worst-off group within each cluster is mitigated as much as possible.
Given that the utilitarian consideration of cluster coherence (one that
classical $k$-means also targets to optimise) would be in apparent tension
with the cluster-level group representativity fairness consideration, we look
to deepen the latter at as little detriment to the former as possible.

Given our intent of mitigating the representativity loss of the worst-off
sensitive group within each cluster, we model our objective function as simply
the aggregate of representativity losses of the worst-off cluster-level group:

(5.7)
$J(\mathcal{C})=\sum_{C\in\mathcal{C}}\max_{S\in\mathcal{S}}\bigg{(}\frac{1}{|C\cap
S|}\sum\limits_{x\in C\cap S}||x-\mu_{C}||^{2}\bigg{)}$

This, as may be noted, captures the fairness ethos espoused by our formulation
of cluster-level group representativity fairness, albeit using significantly
different methodology.

###  5.2 Optimisation Framework

To estimate the parameters – $\mathcal{C}$ and $\\{\ldots\mu_{C}\ldots\\}$ –
in our objective function (Equation 5.7), we follow the same EM-style
iterative procedure as used in the classical $k$-means algorithm that
alternates between the following E and M steps:

  1. 1.

E-step: estimate the cluster assignment for each $x\in X$ keeping the set of
cluster centroids $\\{\ldots\mu_{C}\ldots\\}$ fixed, and

  2. 2.

M-step: estimate the set of cluster centroids $\\{\ldots\mu_{C}\ldots\\}$
keeping the cluster assignment stationary.

The method we propose falls in the category of post-processing methods which
is in line with classical works in fair clustering such as Ahmadian et al [3],
Bera et al [6], Esmaeili et al [17], Kleindessner et al [21], among others. It
operates over the output of a utilitarian clustering algorithm – the classical
$k$-means algorithm; we use the generated cluster centroids as the initial
cluster centroids for our method. We now describe the details of the E and M
steps within our optimisation framework.

###  5.3 E-step: Estimating the Cluster Assignment

Given the set of cluster centroids $\\{\ldots\mu_{C}\ldots\\}$, we need to
assign objects to clusters that minimise the objective function in Equation
5.7. Towards operationalising this, given the complexity of the objective
function, we perturb the existing cluster assignment by greedily reassigning
objects – in round-robin fashion – to new clusters such that the value of the
objective function decreases. Thus, if an object $x\in X$ is reassigned from
cluster $C$ to cluster $C^{\prime}$, the new cluster assignment
$\mathcal{C}^{\prime}$ is given by:

(5.8)
$\mathcal{C}^{\prime}=\mathcal{C}\setminus\big{\\{}C,C^{\prime}\big{\\}}\cup\big{\\{}C\setminus\\{x\\},C^{\prime}\cup\\{x\\}\big{\\}}$

Algorithm 1 outlines our greedy approach. Within each E-step, this entails
trying out $\mathcal{O}(|X|\times k)$ cluster reassignments, $k-1$ per object.
It is notable that the change between $J(\mathcal{C}^{\prime})$ and
$J(\mathcal{C})$ where $\mathcal{C}$ and $\mathcal{C}^{\prime}$ differ in the
membership of a single object can be computed very efficiently without a full
dataset-wide estimation. While we do not include the details of such efficient
computations herein, the computation of such incremental changes is similar in
spirit to what is outlined within Section 4.2.1 in Abraham et al [2].

Algorithm 1 Reassign

1:for all $x\in X$ do

2: for all $C^{\prime}\in\mathcal{C}$ do

3: Obtain the new cluster assignment $\mathcal{C}^{\prime}$ using Equation 5.8

4: if $J(\mathcal{C}^{\prime})<J(\mathcal{C})$ then

5: $\mathcal{C}\leftarrow\mathcal{C}^{\prime}$

###  5.4 M-step: Estimating the Cluster Centroids

We now look at estimating the set of cluster centroids
$\\{\ldots\mu_{C}\ldots\\}$ given the cluster assignment $\mathcal{C}$. Our
goal is to minimise the objective function in Equation 5.7 while keeping the
cluster assignment $\mathcal{C}$ fixed. Since the $\max$ operator is not
differentiable, we use the following differentiable approximation [26]:

(5.9) $\max_{y\in Y}\ g(y)\approx\frac{1}{\phi}\log_{e}\sum_{y\in
Y}\exp{\big{(}\phi\times g(y)\big{)}}$

where $\phi$ is a large enough positive constant that amplifies the
significance of the largest distance. Substituting Equation 5.9 in Equation
5.7 gives us a differentiable approximation for the objective function:

(5.10) $J_{\textit{approx}}=\sum_{C\in\mathcal{C}}\
\frac{1}{\phi}\log_{e}\sum_{S\in\mathcal{S}}\exp{\Big{(}\phi\times\overline{f}(S,C)\Big{)}}$

We model our optimisation steps along the framework of gradient descent where
the intent is to move along the negative gradient. Towards this, we note that
the derivative of $J_{\textit{approx}}$ with respect to $\mu_{C}$ evaluates
to:

(5.11) $\begin{split}\diffp{}{{\mu_{C}}}&J_{\textit{approx}}=\\\
&-2\times\frac{\sum\limits_{S\in\mathcal{S}}\Big{(}w(S,C)\times\frac{1}{|C\cap
S|}\times\sum\limits_{x\in C\cap
S}(x-\mu_{C})\Big{)}}{\sum\limits_{S\in\mathcal{S}}w(S,C)}\end{split}$

where

(5.12) $w(S,C)=\exp{\Big{(}\phi\times\overline{f}(S,C)\Big{)}}$

Much like in the E-step, regularities in the construction of Equation 5.11
allow for efficient incremental gradient computation. Equations 5.11 and 5.12
are together used to update the set of cluster centroids
$\\{\ldots\mu_{C}\ldots\\}$ within the gradient descent framework:

(5.13) $\mu_{C}\leftarrow\mu_{C}-\eta\diffp{}{{\mu_{C}}}J_{\textit{approx}}$

where $\eta\in\mathbb{R}^{+}$ is the learning rate. The M-step is outlined in
Algorithm 2. To summarise, within each M-step, each cluster centroids, chosen
in round-robin fashion, is updated once using the update in Equation 5.13.

Algorithm 2 Update

1:for all $C\in\mathcal{C}$ do

2: Compute $\diffp{}{{\mu_{C}}}J_{\textit{approx}}$ using Equations 5.11 and
5.12

3: Perform a single update for $\mu_{C}$ using Equation 5.13

###  5.5 Stopping Condition

Being initialised using a utilitarian clustering such as one that optimises
for the $k$-means objective, each iteration progressively moves the cluster
assignment away from the utilitarian starting point towards cluster-level
group representativity fairness. This monotonicity makes the choice of the
stopping condition critical to ensure that a good trade-off between the
utilitarianism and fairness is achieved within the eventual cluster
assignment. In other words, the stopping condition for our iterative procedure
is determined by the trade-off between two factors:

  1. 1.

the deterioration in utility, _i.e._ , the classical $k$-means objective,
given that our method starts from the cluster assignment generated by the
classical $k$-means algorithm, and

  2. 2.

the improvement in our objective function towards group fairness.

It may be noted that both the terms above may be computed intrinsically,
_i.e._ , without using any form of external input, and are thus available to
the optimisation approach. This allows us to formulate a contrastive stopping
condition that can choose good trade-off points in the optimisation. At any
iteration, we consider the overall percentage improvement in our fairness
objective over the past ten iterations, and the overall percentage
deterioration in the $k$-means objective across the past ten iterations, and
break out of the iterative loop when the latter exceeds the former. This is
complemented by another stopping condition where we break the iterative
process after 200 iterations, if the contrastive stopping condition was not
reached earlier.

##  6 Empirical Evaluation

We now detail our empirical evaluation. We start by describing the
experimental setup, followed by results and analyses. All underlying research
data and code will be made publicly available upon acceptance.

###  6.1 Experimental Setup

####  6.1.1 Datasets

The datasets used in our empirical evaluation are based on the publicly
available Adult dataset333https://archive.ics.uci.edu/ml/datasets/adult and
CreditCard
dataset444https://archive.ics.uci.edu/ml/datasets/default+of+credit+card+clients
from the UCI repository [15]. Both datasets contain data about humans and
include sensitive information such as race and sex, making them exceedingly
popular in the algorithmic fairness community [11, 23]. In case of the Adult
dataset, we use workclass, education-num, occupation, capital-gain, capital-
loss, hours-per-week as non-sensitive attributes. We do not use fnlwgt (refer
to Le Quy et al [23]) and education (which is the same as education-num). The
two sensitive attributes we consider are sex and race. In case of the
CreditCard dataset, we use LIMIT_BAL, PAY_x (6 attributes), BILL_AMTx (6
attributes), PAY_AMTx (6 attributes) as the non-sensitive attributes. The
sensitive attribute we consider is SEX. The two datasets are processed as
follows. Among the non-sensitive attributes, the continuous ones are
standardised (_i.e._ , zero mean and unit variance) on the same lines as
Ghadiri et al [19], and the categorical ones are one hot encoded. Table 1
contains information about the processed versions of the datasets used in our
evaluation.

dataset | sensitive attribute | non-sensitive attributes | objects
---|---|---|---
name | groups |
Adult | sex | 2 | 26 | 30718
race | 5
CreditCard | SEX | 2 | 77 | 30000
Table 1: Datasets used (after processing)

####  6.1.2 Baselines

We benchmark our method against the classical $k$-means algorithm, which also
forms our initialisation. Ours being a novel fairness formulation that has
been hitherto unexplored in literature, there are no suitable state-of-the-art
baseline clustering methods to compare against. Abbasi et al [1] and Ghadiri
et al [19], being based on group representativity fairness, are related but
are optimised for a different fairness objective; we compare our method
against Ghadiri et al’s Fair-Lloyd [19]. Since the available implementation of
Fair-Lloyd555https://github.com/samirasamadi/SociallyFairKMeans can only
handle binary sensitive attributes, we do not evaluate it on the Adult dataset
with race as the sensitive attribute. We do not compare with Abbasi et al [1]
because it is more suited for the facility location problem rather than the
$k$-means clustering problem.

####  6.1.3 Parameter Configuration

In all experiments, we set the number of clusters to be generated $k$=5. We
run Fair-Lloyd using its default settings. In case of our method, we set
$\phi$=3, and learning rate $\eta$=0.01 for gradient descent. All numbers
reported are averaged over 100 runs with random initial centroids provided to
the classical $k$-means algorithm (whose generated centroids are subsequently
used as the initial centroids of our method). The numbers reported for Fair-
Lloyd are also averaged over 100 runs. Following Ghadiri et al [19]’s setup,
we allow Fair-Lloyd and our method to run for 200 iterations for the purpose
of our empirical evaluation.

###  6.2 Results

####  6.2.1 Fairness vs Utility

dataset |  sensitive attribute | method |  average cluster disparity | % change
---|---|---|---|---
Adult | sex | classical $k$-means | 24.6498 | 0.00%
Fair-Lloyd | 25.0157 | +1.48%
ours | 9.4066 | -61.84%
race | classical $k$-means | 80.8993 | 0.00%
Fair-Lloyd | N/A | N/A
ours | 31.9480 | -60.51%
CreditCard | SEX | classical $k$-means | 21.6800 | 0.00%
Fair-Lloyd | 16.8425 | -22.31%
ours | 4.8128 | -77.80%
Table 2: Average cluster disparity (lower is better). The % change column shows the increase (in red) or decrease (in green) in the average cluster disparity with the classical $k$-means algorithm as the reference.  dataset |  sensitive attribute | method |  $k$-means objective | % change
---|---|---|---|---
Adult | sex | classical $k$-means | 2.0849 | 0.00%
Fair-Lloyd | 2.3548 | +12.95%
ours | 2.5833 | +23.91%
race | classical $k$-means | 2.0860 | 0.00%
Fair-Lloyd | N/A | N/A
ours | 2.3299 | +11.69%
CreditCard | SEX | classical $k$-means | 8.6176 | 0.00%
Fair-Lloyd | 8.5498 | -0.79%
ours | 9.8836 | +14.69%
Table 3: $k$-means objective (lower is better). The % change column shows the
increase/decrease in the $k$-means objective with the classical $k$-means
algorithm as the reference.

Our method optimises for a fairer objective function different from the
classical utilitarian one. It is widely accepted that increase in fairness
almost always causes decrease in utility; it would be of interest to look at
the fairness gains obtained due to our method and the corresponding loss in
utility. Here, we quantify unfairness with our evaluation metric – average
cluster disparity (Equation 4.5), and utility with the $k$-means
objective666We normalise the original $k$-means objective here, _i.e._ ,
divide Equation 3.1 by the number of objects in the dataset.. From Tables 2
and 3, we see that our method is able to significantly reduce the unfairness
(reduction of 61.84%, 60.51%, 77.80% over the classical $k$-means algorithm on
our datasets; 66.72% on average) at the cost of a relatively smaller increase
in the utilitarian objective (increase of 23.91%, 11.69%, 14.69% over the
classical $k$-means algorithm; 16.76% on average). Fair-Lloyd, on the other
hand, does not perform as well on this fairness metric (average reduction of
10%). Evidently, our method outperforms the baselines on our cluster-level
group fairness evaluation metric, _i.e._ , average cluster disparity.

####  6.2.2 Fairness Objectives

dataset |  sensitive attribute | method |  our objective | % change
---|---|---|---|---
Adult | sex | classical $k$-means | 3.1302 | 0.00%
Fair-Lloyd | 3.5403 | +13.10%
ours | 2.1299 | -31.96%
race | classical $k$-means | 4.1148 | 0.00%
Fair-Lloyd | N/A | N/A
ours | 2.7868 | -32.27%
CreditCard | SEX | classical $k$-means | 43.8224 | 0.00%
Fair-Lloyd | 39.6524 | -9.52%
ours | 14.3313 | -67.30%
Table 4: Our objective (lower is better). The % change column shows the increase (in red) or decrease (in green) in our objective with the classical $k$-means algorithm as the reference. dataset |  sensitive attribute | method |  fair $k$-means objective | % change
---|---|---|---|---
Adult | sex | classical $k$-means | 2.2473 | 0.00%
Fair-Lloyd | 2.5453 | +13.26%
ours | 2.7811 | +23.75%
race | classical $k$-means | 2.6185 | 0.00%
Fair-Lloyd | N/A | N/A
ours | 2.8538 | +8.99%
CreditCard | SEX | classical $k$-means | 8.6489 | 0.00%
Fair-Lloyd | 8.5508 | -1.13%
ours | 10.1417 | +17.26%
Table 5: Fair $k$-means objective [1, 19] (lower is better). The % change
column shows the increase/decrease in the fair $k$-means objective with the
classical $k$-means algorithm as the reference.

In the interest of comparing across the two threads of representativity
fairness, we look at how our method compares with the baselines on the two
fairness objectives:  (i) our objective777We normalise our objective here,
_i.e._ , divide Equation 5.7 by the number of clusters $k$. (Equation 5.7),
and  (ii) fair $k$-means objective [1, 19] (Equation 3.2)  . In Table 4, we
see that our method does indeed improve the representativity of the worst-off
sensitive groups (reduction of 31.96%, 32.27%, 67.30% over the classical
$k$-means algorithm; 43.84% on average) which indicates that our method is
moving in the right direction. The Fair-Lloyd baseline does not perform as
well; this is acceptable as it is not designed for cluster-level group
fairness. On the other hand, our method causes the worst-off sensitive group
at the cluster assignment level to have a worse representativity than the
classical $k$-means algorithm, as seen in Table 5. While at first glance this
may seem to be a deficiency of our method, we note that any improvement in the
representativity of this worst-off group would potentially result in another
sensitive group that is the worst-off in some cluster being further
disadvantaged in order to improve the representativity of the former group.
This would be unacceptable in cases where more than one sensitive groups are
historically disadvantaged (_e.g._ , black and American Indians in case of
race) and benefitting one would result in disadvantaging the other.

####  6.2.3 Disparity Trends

Figure 2: Trends in representativity loss $\overline{f}(.,.)$ of the 5
sensitive groups for the Adult dataset with race as the sensitive attribute
(random_state=3, cluster 4) over 100 iterations. Lower is better.

In order to further observe the behaviour of our method, we look at how the
representativity losses of the different sensitive groups vary within a
cluster over iterations. As an illustration, Figure 2 shows the trends in
cluster 4 that was initialised with random_state=3 on the Adult dataset with
race as the sensitive attribute. We notice that over iterations  (i) the
disparity in the representativities of the best-off and worst-off sensitive
groups decreases, and  (ii) the representativity loss of the worst-off
sensitive group decreases  , which is what we expect. Also note in this figure
that the worst-off sensitive group in the cluster changes over iterations;
this is a trend that we observe to generally hold for clusters.

##  7 Conclusion and Future Work

In this paper, we introduced a new formulation of fairness for clustering with
respect to representativity of sensitive groups at the cluster-level. Against
the backdrop of much recent enthusiasm in using representativity-oriented
notions of fairness in clustering, we outlined issues engendered by simplistic
aggregations of representativities across clusters, and how our notion
alleviates such problems. We proposed a new post-processing based clustering
formulation that is able to improve upon existing cluster assignments by
iteratively modifying them towards cluster-level group representativity
fairness. Given the novel form of our fairness notion, we introduced a new
metric that captures disparity between the representativity of groups at the
cluster level. Our experiments on the Adult and CreditCard datasets
demonstrate the effectiveness of our method towards achieving high levels of
cluster-level group representativity fairness at low impact to the popular
utilitarian cluster coherence metric used within $k$-means.

###  7.1 Future Work

One future direction for this work would be to extend our method to
incorporate multiple sensitive attributes together, and also to consider
numeric sensitive attributes such as age, an attribute on which discrimination
is well-understood within healthcare scenarios. Another direction would be to
extend this notion to other clustering paradigms such as hierarchical
clustering and spectral clustering where data models depart significantly from
the relational model we assumed in our present work.

## Acknowledgement

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 945231; and the Department for the Economy in Northern Ireland.
We are grateful for use of the computing resources from the Northern Ireland
High Performance Computing (NI-HPC) service funded by EPSRC (EP/T022175).

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