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We describe the construction of a minimal presentation for the group ofplanar pure braids $\overline{P}_n$ on $n$ strands. The generators of thispresentation are dual to the generators of the cohomology ring of$\overline{P}_n$ found by Y.Baryshnikov while the relations lie in thecommutator subgroup of $\overline{P}_n$. We also construct an explicit(although non-minimal) presentation for the pure cactus group.

A presentation for the planar pure braid group