Preprint Article Version 1 This version is not peer-reviewed

A Symmetry Motivated Link Table

Version 1 : Received: 14 August 2018 / Approved: 15 August 2018 / Online: 15 August 2018 (05:18:48 CEST)

A peer-reviewed article of this Preprint also exists.

Witte, S.; Flanner, M.; Vazquez, M. A Symmetry Motivated Link Table. Symmetry 2018, 10, 604. Witte, S.; Flanner, M.; Vazquez, M. A Symmetry Motivated Link Table. Symmetry 2018, 10, 604.

Journal reference: Symmetry 2018, 10, 604
DOI: 10.3390/sym10110604

Abstract

Proper identification of oriented knots and 2-component links requires a precise link nomenclature. Motivated by questions arising in DNA topology, this study aims to produce a nomenclature unambiguous with respect to link symmetries. For knots, this involves distinguishing a knot type from its mirror image. In the case of 2-component links, there are up to sixteen possible symmetry types for each topology. The study revisits the methods previously used to disambiguate chiral knots and extends them to oriented 2-component links with up to nine crossings. Monte Carlo simulations are used to report on writhe, a geometric indicator of chirality. There are ninety-two prime 2-component links with up to nine crossings. Guided by geometrical data, linking number and the symmetry groups of 2-component links, a canonical link diagram for each link type is proposed. All diagrams but six were unambiguously chosen ( ${8}_{1}^{2}5$ , ${9}_{5}^{2}$ , ${9}_{3}^{2}4$ , ${9}_{3}^{2}5$ , ${9}_{3}^{2}9$ , and ${9}_{4}^{2}1$ ). We include complete tables for prime knots with up to ten crossings and prime links with up to nine crossings. We also prove a result on the behavior of the writhe under local lattice moves.

Subject Areas

writhe; chirality; nomenclature; link symmetries; link table; knot table; lattice polygons; DNA topology

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