Mathematics

Fix an edge-length vector $r=(r_1,\ldots,r_n)$ of an $n$-gon lying in the Euclidean plane. The moduli space of $n$-gons, $M_r$, is the space of all polygons with distinguished vertices and edge-length vector $r$, modulo orientation preserving isometries of the plane. The topology of $M_r$ depends on the combinatorics of $r$. Kapovich and Millson in 1996 produced a certain polytopal complex $D_n$ which parametrizes the topologies that may arise in $M_r$, and used this complex to list the possible topologies of the moduli space of $n$-gons for $n

By factoring out the symmetries of $D_n$, and with the use of modern computational geometry, I will improve the results of Kapovich and Millson in the case $n=6$, and provide results in the case $n=7$.