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Gromov distortion measures the ratio between distance along the knot and distance in the ambient space. No nontrivial knot class has its Gromov distortion invariant calculated because of its complexity. While upper bounds on numerical knot invariants are easily given by examples, the Gromov distortion calculation of even the most contrived diagrams has required significant proof.

Vertex distortion of lattice knots was introduced as a combinatorial approach to understand the enigmatic Gromov distortion of smooth knots. We show that vertex distortion detects the unknot and that vertex distortion is bounded above and below be a constant multiple of Gromov distortion. This allows for the port of Pardon’s well-known result on Gromov distortion to vertex distortion.

Lattice knots are used in DNA Topology since arbitrary smooth knots do not model the rigidity of the molecule well. We give a vertex distortion calculator that can detect the unknot, if a small value is returned.