Mathematics & Computer Science, Geometry & Topology; writhe; chirality; nomenclature; link symmetries; link table; knot table; lattice polygons; DNA topology
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.