The topological organization of the turtle cranium is constrained and conserved over long evolutionary timescales.

The cranium of turtles (Testudines) is characterized by the secondary reduction of temporal fenestrae and loss of cranial joints (i.e., characteristics of anapsid, akinetic skulls). Evolution and ontogeny of the turtle cranium are associated with shape changes. Cranial shape variation among Testudines can partially be explained by dietary and functional adaptations (neck retraction), but it is unclear if cranial topology shows similar ecomorphological signal, or if it is decoupled from shape evolution. We assess the topological arrangement of cranial bones (i.e., number, relative positioning, connections), using anatomical network analysis. Non-shelled stem turtles have similar cranial arrangements to archosauromorph outgroups. Shelled turtles (Testudinata) evolve a unique cranial organization that is associated with bone losses (e.g., supratemporal, lacrimal, ectopterygoid) and an increase in complexity (i.e., densely and highly interconnected skulls with low path lengths between bones), resulting from the closure of skull openings and establishment of unusual connections such as a parietal-pterygoid contact in the secondary braincase. Topological changes evolutionarily predate many shape changes. Topological variation and taxonomic morphospace discrimination among crown turtles are low, indicating that cranial topology may be constrained. Observed variation results from repeated losses of nonintegral bones (i.e., premaxilla, nasal, epipterygoid, quadratojugal), and changes in temporal emarginations and palate construction. We observe only minor ontogenetic changes. Topology is not influenced by diet and habitat, contrasting cranial shape. Our results indicate that turtles have a unique cranial topology among reptiles that is conserved after its initial establishment, and shows that cranial topology and shape have different evolutionary histories.