A Symmetry Motivated Link Table

Shawn Witte,Mariel Vazquez,Michelle Flanner

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 link type. 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, canonical link diagrams for all but five link types (9 5 2, 9 34 2, 9 35 2, 9 39 2, and 9 41 2) are proposed. 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.

Highlights

The unambiguous identification of links that are not topologically equivalent is of utmost relevance when studying links in a natural setting
We choose the standard link diagram L representative of the set L as the one corresponding to an isotopy class where the sum of self-writhes and linking number are most positive, with components labeled in order of decreasing self-writhe. We argue that this is a natural approach to choose the standard link diagram since writhe and linking number are very closely related to chirality, and since we find that component self-writhe is related to exchange symmetries
Independent ensembles of linked lattice polygons were obtained as described in Sections 4.1 and 6

Summary

Introduction

The unambiguous identification of links that are not topologically equivalent is of utmost relevance when studying links in a natural setting. Of special interest in the field of DNA topology is the action of enzymatic processes that produce DNA links In this setting, one needs proper distinction between a link and its mirror image, or between two links related by reflection, orientation reversal, or component relabeling. Enzymes in the family of type II topoisomerases pass a segment of a DNA molecule through another introducing crossing changes (Figure 1b). Another class of enzymes, site-specific recombinases, bind to two DNA segments, cleave and reconnect the ends (Figure 1a). A mislabeling of the component orientation, or a mistaken chirality of the product can have severe implications on the mechanistic study See more on this topic at the end of this section and in

Objectives

Methods

Results

Conclusion