Witte, S.; Flanner, M.; Vazquez, M. A Symmetry Motivated Link Table. Symmetry2018, 10, 604.
Witte, S.; Flanner, M.; Vazquez, M. A Symmetry Motivated Link Table. Symmetry 2018, 10, 604.
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 ( , , , , , and ). 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.
writhe; chirality; nomenclature; link symmetries; link table; knot table; lattice polygons; DNA topology
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.