Abstract:
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.
Key words and phrases:knot theory, Gordian graph, Gordian complex, crossing change, non-splittability, satellite knots, incompressible tori.
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