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Research ArticleNo Access

Vertex distortion detects the unknot

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

E-mail Address: marion.campisi@sjsu.edu

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https://orcid.org/0000-0001-8583-8563

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

E-mail Address: nccazet@ucdavis.edu

Corresponding author.

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

E-mail Address: dcrncevi@u.rochester.edu

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

E-mail Address: fellmt@rpi.edu

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Phillip Kessler

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

E-mail Address: kesslerp@uci.edu

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

E-mail Address: nri237@uky.edu

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Vatsal Srivastava

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

E-mail Address: 18d180029@iitb.ac.in

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, and

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

E-mail Address: luistorres@utexas.edu

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https://doi.org/10.1142/S0218216522500742Cited by:0 (Source: Crossref)

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.

Keywords:

AMSC: 57K10, 57M50

References

1. R. Blair, M. Campisi, S. A. Taylor and M. Tomova, Distortion and the bridge distance of knots, J. Topol. 13(2) (2020) 669–682. Crossref, Web of Science, Google Scholar
2. M. Campisi and N. Cazet, Vertex distortion of lattice knots, J. Knot Theory Ramifications 30(7) (2021) 2150043. Link, Web of Science, Google Scholar
3. E. Denne and J. M. Sullivan, The distortion of a knotted curve, Proc. Amer. Math. Soc. 137(3) (2009) 1139–1148. Crossref, Web of Science, Google Scholar
4. M. Gromov, Filling Riemannian manifolds, J. Differential Geom. 18(1) (1983) 1–147. Crossref, Web of Science, Google Scholar
5. J. Pardon, On the distortion of knots on embedded surfaces, Ann. Math. 174(1) (2011) 637–646. Crossref, Web of Science, Google Scholar

Journal cover image

Vol. 31, No. 11

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Downloaded 45 times

History

Received 5 January 2022

Revised 19 August 2022

Accepted 20 August 2022

Published: 11 October 2022

Keywords