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Link Nomenclature, Random Grid Diagrams, and Markov Chain Methods in Knot Theory

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Link Nomenclature, Random Grid Diagrams, and Markov Chain Methods in Knot Theory Link Nomenclature, Random Grid Diagrams, and Markov Chain Methods in Knot Theory By SHAWN L. WITTE DISSERTATION Submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in MATHEMATICS in the OFFICE OF GRADUATE STUDIES of the UNIVERSITY OF CALIFORNIA DAVIS Approved: Mariel Vazquez, Chair Javier Arsuaga Eric Babson Committee in Charge 2019 i Contents Abstract iv Acknowledgments v Chapter 1. Introduction 1 Chapter 2. Definitions and Background 4 2.1. Combinatorics 4 2.2. Computation 4 2.3. Knot Theory 5 2.4. Markov Chains 25 2.5. Questions and Conjectures 34 Chapter 3. Link Nomenclature 39 3.1. Symmetries, Writhe and Linking Number 39 3.2. Canonical Isotopy Class 40 3.3. Numerical Writhe Results for 2-Component Links 42 3.4. Writhe and Minimum Step Conformations 46 Chapter 4. Markov Chain Methods for Lattice Links 48 4.1. Link-BFACF 49 4.2. Wang-Landau for Lattice Links 55 Chapter 5. Combinatorics and Writhe of Grid Diagrams 59 5.1. Combinatorics of Grid Diagrams 59 5.2. Grids and Writhe 65 Chapter 6. The Frisch-Wasserman-Delbr¨uck Conjecture(FWD) for Grid Diagrams 72 6.1. Proof of the FWD Conjecture for Grid Diagrams 73 Chapter 7. Random Grid Diagrams 85 ii 7.1. Uniformly Random n × n Grid Diagrams 85 7.2. Markov Chain Algorithms for Grid Diagrams 86 7.3. Numerical Results for Grid Diagrams 96 Chapter 8. Conclusion 102 Appendix A. Index of Notation 104 Appendix B. Link Nomenclature Data 106 Appendix C. Oriented Labeled Link Table 121 Bibliography 126 iii Abstract With the support of data from BFACF Markov chain Monte Carlo sampling, conjectures are formed stating that the mean writhe of 2-component lattice links with fixed length and knot type are bounded as length increases. Further, the self-writhe of the components exhibit a similar behav- ior. It is further conjectured that these writhes sufficiently distinguish reflections and component relabelings of links lacking those symmetries. Guided by these conjectures and using the writhe data obtained, a set of canonical isotopy classes are proposed. Further refinements of the BFACF algorithm are also presented, including a modification into a Wang-Landau algorithm. The behav- ior of writhe under BFACF moves is proven to be bounded and determined precisely by the local geometry around the edge where the move is being performed. The combinatorics of grid diagrams are explored. In particular, n×n grid diagrams representing c-component links and the total number of n × n grid diagrams are enumerated precisely. A lower bound is provided for the number of n × n grid diagrams representing a specific knot type. It is proven that the probability that an n × n grid diagram of a knot chosen uniformly at random being of knot type K decays to 0 as n ! 1, i.e. the Frisch-Wasserman-Delbr¨uck (FWD) conjecture. This proof provides an upper bound to the number of n × n grid diagrams representing a specific knot type. It is also shown that the change in writhe from performing knot-preserving grid moves is bounded and can be determined precisely from only a few entries related to where the move is being performed. Two Markov chain algorithms for randomizing grid diagrams of fixed knot type are presented, one of which is a Wang-Landau algorithm. The Wang-Landau algorithm is used to explore the decay of the knotting probability and shows that the upper bound on the number of n × n grid diagrams representing a specific knot type given by the proof of the Frisch-Wasserman-Delbr¨uck is not tight. Using this data, it is conjectured that this decay is exponential or faster. The Wang- Landau algorithm is also used to explore the mean writhe of n × n grid diagrams representing a fixed knot type. These results are used to conjecture that the mean writhe of n × n grid diagrams is bounded as n ! 1 and can be used to distinguish a chiral knot and its mirror image. iv Acknowledgments I'd like to acknowledge the many people who made this dissertation possible. Reuben Brasher helped with getting the BFACF software running and for general knowledge about the BFACF algorithm. Rob Scharein provided a copy of KnotPlot and was available for support in using that software. Robert Stolz and I had many conversations regarding BFACF, correlation, and Markov chain sampling. Michelle Flanner collaborated on the link nomenclature work and provided invaluable technical assistance at all levels and all stages of my work. Zihao Zhu did a large amount of work in implementing a version of Wang-Landau for lattice links. Tamara Christiani was always a ready sounding board for any of my ideas. Maxime Puokam was available to consult with on statistical matters. Chris Soteros provided a constant stream of ideas. Andrew Rechnitzer presented a workshop on the Wang-Landau algorithm which inspired all of the work done from that point forward. Chaim Even-Zohar pointed out the possible extension of his FWD proof from petal diagrams to grid diagrams. Marc Culler created Gridlink from which the original versions of my grid diagram codebase were based and from which I obtained my initial conformations for each knot type. My Committee members Javier Arsuaga and Eric Babson provided useful feedback on this work. My advisor, Mariel Vazquez, provided guidance through the whole process. I'd also like to thank the many artists who provided the soundtrack to which I wrote this dissertation: Sammus, Holly Herndon, Lingua Ignota, Carly Rae Jepson, Mannequin Pussy, and many many others. v CHAPTER 1 Introduction Of primary importance in knot theory is determining the equivalence or inequivalence of knots and links. This is usually done through the use of topological invariants (Section 2.3.6). Many of these invariants do not distinguish knots or links that differ only by a mirroring, orientation change, or relabeling of components, i.e. different isotopy classes of a link. This is okay for a broad study of knots and links, but these can be crucial considerations for particular applications. Take, for example, double stranded deoxyribonucleic acid (dsDNA). In B-form dsDNA, the two sugar phosphate backbones wrap around each other forming a double helix that twists in a right- handed fashion [BBM+05]. Bacterial DNA is circular, meaning that the strands of the double helix form a 2-component link. For the cell to undergo mitosis, this DNA must be replicated. This operation is performed by enzymes and proteins in the cell which unzip the DNA into two complementary strands which are then synthesized back into double stranded DNA. If this operation is performed as stated, then the two new DNA molecules will necessarily be linked in the same fashion as the backbones of the original helix, preventing them from being pulled into separate daughter cells. This means that there must be unlinking machinery in the cell which resolve the linked DNA into two unlinked molecules. The machinery comes in the form of recombinases and type II topoisomerases which perform operations equivalent to diagrammatic smoothings and crossing changes (as in the skein relations inSection 2.3.6.3), respectively. It has been observed that these mechanisms unlink the DNA more efficiently than a random process would [Ryb97]. It has also been shown that topo IV, a type II topoisomerase, has a chiral bias [Cri00]. Several studies have been focused on the study of unlinking pathways, i.e. the series of intermediate links and/or knots that a pair of DNA molecules pass through as they are unlinked [SYB+17, BI15, VCS05]. In those studies, it is important to consider the specific isotopy classes of knots and links along the pathway, as a chiral bias could affect the appearance of links vs. their mirror images. To communicate knot and link types of low crossing number, one usually uses Rolfsen's table [Rol76] (preferably a version without the infamous \Perko pair" [Per14]). This table is sufficient 1 when isotopy classes are not considered, and can be used as a reference point for the mirror of a knot or link. However, this table omits orientations and does not distinguish the components of links in any meaningful way. Further, for links which are not equivalent to their mirror images (chiral links), there is no particular mathematical reasoning for which isotopy classes are presented in the table and which are not. Before exploring the questions raised by these facts, Chapter 2 provides the necessary back- ground and definitions used throughout this work. A comprehensive and rigorous explanation of the questions examined here can be found in Section 2.5. Given the length and breadth of the work presented here, an index of notation is provided in Appendix A. Markov chain Monte Carlo experiments have provided evidence that low-crossing chiral knots can be classified into \positive" and \negative" chiralities based on the sign of the mean writhe (defined in section 2.3.7) of fixed length lattice knots of that knot type [PDS+11, SIA+09]. In Chapter 3, similar methods are used to examine the writhe of 2-component links. The results are used to select a canonical isotopy class for each prime link with crossing number 9 or less. A diagram for each canonical isotopy class can be found in Appendix C. The data used to obtain these isotopy classes was found by sampling random conformations of the links represented by unions of self-avoiding polygons embedded in the simple cubic lattice (Section 2.3.5). The samples were obtained from Markov chain Monte Carlo sampling using the BFACF algorithm (Section 2.4.4).
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