The topology of $SLE_{\kappa}$ is random for $\kappa >4$

Abstract

We study the topology of $SLE$ curves for $\kappa > 4$. More precisely, we show that, a.s., there is no homeomorphism $\Phi: \overline{\mathbb{H}} \rightarrow \overline{\mathbb{H}}$, taking the range of one independent $SLE$ curve to another for $\kappa \in (4,8)$. Furthermore, we extend the result to $\kappa \geq 8$ by showing that there is no homeomorphism taking one $SLE$ curve to another, when viewed as curves modulo parametrization.