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\begin{document}

Finally \DIFdelbegin \DIFdel{, we have the }\DIFdelend \DIFaddbegin \DIFadd{we have Burkhardt's }\DIFaddend $g-1$ factor swaps.  These are \DIFdelbegin \DIFdel{easily visualized as ``handle swaps,''}\DIFdelend \DIFaddbegin \DIFadd{obtained as the 
$\Psi$-images of handle swaps.  The $i{\rm th}$ }{\em \DIFadd{handle 
swap}}\DIFaddend \index{handle swap} \DIFaddbegin \DIFadd{$h_i$ for $1\leq i\leq g-1$ is easily visualized 
(see Figure~\ref{figure:burkhardt}), }\DIFaddend but we can also write \DIFdelbegin \DIFdel{them as products }\DIFdelend \DIFaddbegin \DIFadd{it as a product 
}\DIFaddend of Dehn twists\DIFdelbegin \DIFdel{, for example: 
}\[\DIFdel{ (T_{a_2}T_{b_2} T_{a_2+b_1} T_{a_1} T_{b_1})^3, }\]
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\DIFdel{where $a_2+b_1$ }\DIFdelend \DIFaddbegin \DIFadd{: 
}

$$\DIFadd{h_i=(T_{a_{i+1}}T_{b_{i+1}} T_{d_i}T_{a_i}T_{b_i})^3}$$ 
\DIFadd{where $d_i$ }\DIFaddend is a simple closed curve \DIFdelbegin \DIFdel{representing $[a_2]+[b_1]$, or in general $(T_{a_{i+1}}T_{b_{i+1}} T_{a_{i+1}+b_i}T_{a_i}T_{b_i})^3$, where 
$a_{i+1}+b_i$ represents }\DIFdelend \DIFaddbegin \DIFadd{that in the homology class 
}\DIFaddend $[a_{i+1}]+[b_i]$\DIFdelbegin \DIFdel{and $1 \leq i \leq g-1$}\DIFdelend .

\end{document}
