Multicorns are not Path Connected

The “multicorn” is the connectedness locus of unicritical antiholomorphic polynomials $z\mapsto \bar{r}{z}^d+c$; the special case $d=2$ was named “tricorn” by Milnor. It appears as a natural local configuration in spaces of real cubic polynomials. We prove that no multicorn for $d\ge 2$ is pathwise connected, confirming a classical prediction based on numerical observations.