Vertex distortion detects the unknot

Author:

Campisi Marion1,Cazet Nicholas2ORCID,Crncevic David3,Fellman Tasha4,Kessler Phillip5,Rieke Nikolas6,Srivastava Vatsal7,Torres Luis8

Affiliation:

1. San Jóse State University, 1 Washington Square, San Jóse, CA 95192, USA

2. University of California at Davis, 1 Shields Avenue, Davis, CA 95616, USA

3. University of Rochester, 500 Joseph C Wilson Blvd, Rochester, NY 14627, USA

4. Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180, USA

5. University of California at Irvine, 510 E Peltason Dr, Irvine, CA 92697, USA

6. University of Kentucky, 719 Patterson Office Tower, Lexington, KY 40506, USA

7. Indian Institute of Technology (IIT) Bombay, Powai, Mumbai 400 076, Maharashtra, India

8. The University of Texas at Austin, 110 Inner Campus Drive, Austin, TX 78705, USA

Abstract

The first two authors introduced vertex distortion of lattice knots and showed that the vertex distortion of the unknot is 1. It was conjectured that the vertex distortion of a knot class is 1 if and only if it is trivial. We use Denne and Sullivan’s lower bound on Gromov distortion to bound the vertex distortion of non-trivial lattice knots. This bounding allows us to conclude that a knot class has vertex distortion 1 if and only if it is trivial. We also show that vertex distortion does not have a universal upper bound and provide a vertex distortion calculator.

Publisher

World Scientific Pub Co Pte Ltd

Subject

Algebra and Number Theory

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