Infinite midway cliques in the Gordian graph

Inspired by results of Hirasawa, Uchida, and Baader, we reveal a new geometric pattern in the Gordian complex of knots. We prove that for any two vertices at Gordian distance 2, the intersection of their 1-neighborhoods contains an infinite-dimensional simplex. The proof relies on a new geometric sufficient condition of the non-splittability of links, based on an iterative construction of gropes from unknotted one-holed tori. As a corollary, the Gordian graph remains connected after removing any induced locally finite subgraph.