CaTherine wheels from trees and Liouville quantum gravity

A CaTherine wheel is a space-filling curve $f:S^1\to S^2$ such that for every closed interval $J\subset S^1$, $f(J)$ is homeomorphic to a closed disk and $f(\mathrm{\partial }J)$ is contained in $\mathrm{\partial }f(J)$. A CaTherine wheel gives rise to a pair of disjoint, dense topological trees in $S^2$ which roughly speaking lie to the left and right of $f$. We give necessary and sufficient conditions for a topological tree in $S^2$ to arise as one of these trees for some CaTherine wheel $f$. We apply this result to show that there is a unique CaTherine wheel corresponding to the geodesic tree rooted at $\mathrm{\infty }$ for the $\gamma$-Liouville quantum gravity (LQG) metric, for $\gamma \in (0,2)$. In other words, we construct the space-filling curve which is the contour exploration of the LQG geodesic tree.