November 2018
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74 Reads
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3 Citations
Symmetry
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.
![(A) Under the assumption that each reconnection step strictly reduces the number of crossings of the substrate, in Shimokawa et al.¹⁰ we showed that there is a unique unlinking pathway starting at a 2m-crossing replication link. In E. coli a replication link is a 2m-cat with parallel dif sites⁶, and this pathway predicts the first product to be a (2m−1)1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{(2}m-\mathrm{1)}}_{1}$$\end{document} knot with two dif sites in direct repeats. Two sites along a knotted chain are in direct repeats if they induce the same orientation into the knot. Replication links are 2m-crossing right-handed torus links with parallel sites (mathematical notation: (2m)12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{(2}m)}_{1}^{2}$$\end{document}). The pathway in the figure illustrates, for m = 6, the only unlinking pathway starting at the parallel 2m-cat under the assumption that each reconnection step strictly reduces the minimal crossing number. All the intermediate topologies are torus links (2m)12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{(2}m)}_{1}^{2}$$\end{document} or torus knots (2m−1)1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{(2}m-\mathrm{1)}}_{1}$$\end{document} with two reconnection sites in direct repeats as in the figure. (B) One reconnection step: here the cleavage regions of the reconnection sites on a 612\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${6}_{1}^{2}$$\end{document} link are brought together to form a synapse (shown as a ball enclosing two strings). The synapse is modeled mathematically as a 2-string tangle. In the case of XerCD site-specific recombination, the strings in the tangle contain the core regions of the dif sites (indicated by two arrows in a tangle P representing two very short segments of double-stranded DNA which physically behave as two almost straight strings) and any bound DNA which does not change during recombination (gray shaded region). Any interesting geometrical or topological complexity of the substrate is captured mathematically as an outside tangle O that remains constant during reconnection. Before strand cleavage, the substrate is modeled by the tangle equation N(O+P)=612\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N(O+P)={6}_{1}^{2}$$\end{document}. The local reconnection is modeled by tangle surgery where P is replaced with R, yielding a product represented as N(O+R)=K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N(O+R)=K$$\end{document}, where K is a knot with two directly repeated sites. (C) Local reconnection is a simple event which can be modeled as a band surgery, where P = (0) is replaced with a tangle R = (w, 0) enclosing a vertical row of w twists, for some integer w. The rational tangle notation (or Conway notation) for such vertical tangle is R = (w, 0). In the case when w=±1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w=\pm 1$$\end{document} the notation simplifies to R=(±1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R=(\pm \mathrm{1)}$$\end{document}. In the simplest cases, P = (0) with sites in parallel alignment goes to R=(±1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R=(\pm \mathrm{1)}$$\end{document}, and P = (0) with sites in anti-parallel alignment goes to R=(0,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R=\mathrm{(0,}\,\mathrm{0)}$$\end{document} as illustrated in the figure.](https://www.researchgate.net/publication/320108755/figure/fig3/AS:959186234011664@1605699227210/A-Under-the-assumption-that-each-reconnection-step-strictly-reduces-the-number-of_Q320.jpg)
![Quantification of the time-course experiments⁹. The gel presented in Fig. 1B in Grainge et al.⁹ showed a time course of unlinking by XerCD-dif-FtsK50C at 25 °C of newly replicated plasmids containing dif sites. Line scans of the gel were previously published¹⁰. In this figure each topological class is shown as a separate series of points with linear interpolation. The caption assumes the bands observed correspond to the topologies expected from a substrate composed of replication links, i.e. 2m-crossing links (e.g. 2m-cats), and some of the corresponding knotted intermediates (open circle or 01, 31, 51). “Unlink” corresponds to the two unlinked components in monomeric state (topology type 012\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${0}_{1}^{2}$$\end{document}), and “Unknot” corresponds to the dimeric unknot (01). The quantification clearly illustrates the reduction of replication links by XerCD-FtsK site-specific recombination at dif sites. The complexity of the data is also evident, with the relative proportions of all the different topologies fluctuating from one step to the next, thus obscuring the signal.](https://www.researchgate.net/publication/320108755/figure/fig4/AS:959186233999366@1605699227509/Quantification-of-the-time-course-experiments-The-gel-presented-in-Fig1B-in-Grainge-et_Q320.jpg)
