DiskScissors: Cutting Arbitrary‐Topology Solids for Bijective Mapping <jats:title>Abstract</jats:title> <jats:p> An algorithm for cutting solid objects in a topology‐controlled manner is presented. Concretely, given a loop on the object boundary, a disk‐topology cut surface bounded by the loop is constructed in the interior. In contrast to various previous approaches, both disk topology and conformance to the prescribed loop are ensured by construction, while supporting not only contractible but also incontractible loops on the boundaries of manifold objects of higher genus and arbitrary non‐trivial topology. We describe an implementation of this algorithm in the discrete setting, with triangle mesh cut surfaces embedded in tetrahedral mesh objects. Making use of this novel cutting algorithm, we describe a method for the reliable construction of bijective volumetric maps between solid objects, demonstrating the algorithm's utility. This mapping method overcomes restrictions of the state of the art to topological balls, extending coverage to objects of arbitrary genus, specifically so‐called <jats:italic>1</jats:italic> ‐handlebodies. </jats:p> Wiley