We consider the Schramm-Loewner evolution (SLEκ) with κ = 4, the critical value of κ > 0 at or below which SLEκ is a simple curve and above which it is self-intersecting. We show that the range of an SLE4 curve is a.s. conformally removable. Such curves arise as the conformal welding of a pair of independent critical (γ = 2) Liouville quantum gravity (LQG) surfaces along their boundaries and our result implies that this conformal welding is unique. In order to establish this result, we give a new sufficient condition for a set X ⊆ ℂ to be conformally removable which applies in the case that X is not necessarily the boundary of a simply connected domain. We will also describe how this theorem can be applied to obtain the conformal removability of the SLEκ curves for κ ∈ (4,8) in the case that the adjacency graph of connected components of the complement is a.s. connected. This talk will assume no prior knowledge of SLE or LQG.
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