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We study the problem of quasisymmetrically embedding spaces homeomorphic tothe Sierpi\'nski carpet into the plane. In the case of so called dyadic slitcarpets, several characterizations are obtained. One characterization is interms of a Transboundary Loewner Property (TLP) which is a transboundaryanalogue of the Loewner property of Heinonen and Koskela. We show that a dyadicslit carpet can be quasisymmetrically embedded into the plane if and only if itis TLP. Moreover, every dyadic slit carpet $X$ can be associated to a"pillowcase sphere" $\widehat{X}$ which is a metric space homeomorphic to thesphere $\mathbb{S}^2$. We show that $X$ quasisymmetrically embeds into theplane if and only if $\widehat{X}$ is quasisymmetric to $\mathbb{S}^2$ if andonly if $\widehat{X}$ is Ahlfors $2$-regular.

Quasisymmetric embeddings of slit Sierpi\'nski carpets