DOCSLIB.ORG

Mia Nguyen University of Nebraska-Lincoln, Department of Mathematics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mia Nguyen University of Nebraska-Lincoln, Department of Mathematics The Hexagonal Lattice Number of the Figure Eight is 11 2019 Nebraska Conference for Undergraduate Women in Mathematics Mia Nguyen University of Nebraska-Lincoln, Department of Mathematics Quick review of knot theory are able to make some estimates about the stick number of some simple knots. From the cubic model of the knot, we project it Knot theory is a branch of topology that studies three- onto the xyw-plane and transform it to hexagonal lattice. For dimensional manifolds. A mathematical knot is a closed curve the conversion, the angle of 30o at each corner and the minimal that is embedded in 3-dimensional Euclidean space. Two or number of sticks are prominent. more knots combined together are considered as a link. It is not obvious to determine if 2 given knots are equivalent to each other or not. Proving stick number of the figure eight is Definition of hexagonal lattice 11 Hexagonal lattice includes points such that an equilateral trian- gle is formed by every 3 nearby points. There are 4 orientations associated with the lattice, one goes up, one goes to the right, and 2 oblique axes. The simple hexagonal lattice is defined as the point lattice where x = <1, 0, 0>, y = <1/2, 3/2, 0>, w = <0, 0, 1>, and z = y - x. Figure 5: The 52 knot and its projection in the simple hexagonal lattice. The x-stick, y-, z-, and w-sticks are straight line segments that are parallel to directions of x, y, z, and w. Stick number is the In Figure 5, jP jx = 2; jP jy = 4; jP jz = 3; jP jw = 5, and smallest number of edges needed to form a knot. There are not jP j = 14. many knots that their stick numbers are achieved exactly be- cause it is hard to determine if we reach the minimal number of stick. jP jx; jP jy; jP jz, and jP jw will denote the number of x- sticks, y-sticks, z-sticks, and w-sticks, respectively, and jP j will Figure 2: The 41 knot and its minimal stick conformation in the simple cubic show the total number of sticks. lattice. Figure 6: The 61 knot and its projection in the simple hexagonal lattice. In Figure 6, jP jx = 3; jP jy = 2; jP jz = 4; jP jw = 7, and jP j = 16. Figure 3: The projection of the figure eight in the simple hexagonal lattice. Figure 1: The directions of axes in the simple hexagonal lattice. In Figure 3, jP jx = 2; jP jy = 2; jP jz = 3; jP jw = 4; and jP j = 11. Theorem about stick number Theorem 1 Conclusions For any knot K in the simple hexagonal lattice, s[K] ≥ 5b[K]. We know that lower bound for stick numbers of any knot in Knots with the bridge index equal to 2 have different stick num- the simple hexagonal lattice is equal or greater than 5b[K] where bers. The figure eight and trefoil are the only 2 knots that we b[K] is the bridge number of the knot K (Theorem 1). Since the could prove the minimal number of sticks is 11. Figure 7: The 62 knot and its projection in the simple hexagonal lattice. bridge number of trefoil is 2 with 2 local minima and 2 local maxima, its stick number should be at least 10. In Figure 7, jP jx = 3; jP jy = 3; jP jz = 2; jP jw = 7, and jP j = 15. Theorem 2 Further work In the simple hexagonal lattice, the stick number of any non- References trivial knot is at least 11. [1] Ryan Bailey, Hans Chaumont, Melanie Dennis, Jennifer We need at least 10 sticks to construct any nontrivial knot (The- McLoud-Mann, Elise McMahon, Sara Melvin, and Geoffrey orem 1). However, it was shown that any knot built with 10 Schuette. Stick numbers in the simple hexagonal lattice. In- sticks is trivial. That leads to a conjectured new lower bound for volve, a Journal of Mathematics, pages 503–512, 2015. stick number. [2] C. E. Mann, J. C. McLoud-Mann, and D. P. Milan. The stick number for the simple hexagonal lattice. J. Knot Theory Conjecture Ramifications, pages 14–21, 2012. For any knot K in the simple hexagonal lattice, Acknowledgements I gratefully acknowledge the very helpful discussions pro- Figure 4: The 51 knot and its projection in the simple hexagonal lattice. s[K] ≥ 5b[K] + 1 vided by Professor Alex Zupan, my research supervisor, for In my research project, I proved that the hexagonal lattice num- In Figure 4, jP jx = 2; jP jy = 3; jP jz = 2; jP jw = 6, and his enthusiasm, valuable support, and encouragement of this ber of the figure eight is also 11. Based on bridge number, we jP j = 13. research work..
Load more

Recommended publications
  • The Khovanov Homology of Rational Tangles
    The Khovanov Homology of Rational Tangles Benjamin Thompson A thesis submitted for the degree of Bachelor of Philosophy with Honours in Pure Mathematics of The Australian National University October, 2016 Dedicated to my family. Even though they’ll never read it. “To feel fulfilled, you must first have a goal that needs fulfilling.” Hidetaka Miyazaki, Edge (280) “Sleep is good. And books are better.” (Tyrion) George R. R. Martin, A Clash of Kings “Let’s love ourselves then we can’t fail to make a better situation.” Lauryn Hill, Everything is Everything iv Declaration Except where otherwise stated, this thesis is my own work prepared under the supervision of Scott Morrison. Benjamin Thompson October, 2016 v vi Acknowledgements What a ride. Above all, I would like to thank my supervisor, Scott Morrison. This thesis would not have been written without your unflagging support, sublime feedback and sage advice. My thesis would have likely consisted only of uninspired exposition had you not provided a plethora of interesting potential topics at the start, and its overall polish would have likely diminished had you not kept me on track right to the end. You went above and beyond what I expected from a supervisor, and as a result I’ve had the busiest, but also best, year of my life so far. I must also extend a huge thanks to Tony Licata for working with me throughout the year too; hopefully we can figure out what’s really going on with the bigradings! So many people to thank, so little time. I thank Joan Licata for agreeing to run a Knot Theory course all those years ago.
    [Show full text]
  • Knots, Links, Spatial Graphs, and Algebraic Invariants
    689 Knots, Links, Spatial Graphs, and Algebraic Invariants AMS Special Session on Algebraic and Combinatorial Structures in Knot Theory AMS Special Session on Spatial Graphs October 24–25, 2015 California State University, Fullerton, CA Erica Flapan Allison Henrich Aaron Kaestner Sam Nelson Editors American Mathematical Society 689 Knots, Links, Spatial Graphs, and Algebraic Invariants AMS Special Session on Algebraic and Combinatorial Structures in Knot Theory AMS Special Session on Spatial Graphs October 24–25, 2015 California State University, Fullerton, CA Erica Flapan Allison Henrich Aaron Kaestner Sam Nelson Editors American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Catherine Yan 2010 Mathematics Subject Classification. Primary 05C10, 57M15, 57M25, 57M27. Library of Congress Cataloging-in-Publication Data Names: Flapan, Erica, 1956- editor. Title: Knots, links, spatial graphs, and algebraic invariants : AMS special session on algebraic and combinatorial structures in knot theory, October 24-25, 2015, California State University, Fullerton, CA : AMS special session on spatial graphs, October 24-25, 2015, California State University, Fullerton, CA / Erica Flapan [and three others], editors. Description: Providence, Rhode Island : American Mathematical Society, [2017] | Series: Con- temporary mathematics ; volume 689 | Includes bibliographical references. Identifiers: LCCN 2016042011 | ISBN 9781470428471 (alk. paper) Subjects: LCSH: Knot theory–Congresses. | Link theory–Congresses. | Graph theory–Congresses. | Invariants–Congresses. | AMS: Combinatorics – Graph theory – Planar graphs; geometric and topological aspects of graph theory. msc | Manifolds and cell complexes – Low-dimensional topology – Relations with graph theory. msc | Manifolds and cell complexes – Low-dimensional topology – Knots and links in S3.msc| Manifolds and cell complexes – Low-dimensional topology – Invariants of knots and 3-manifolds.
    [Show full text]
  • Planar and Spherical Stick Indices of Knots
    PLANAR AND SPHERICAL STICK INDICES OF KNOTS COLIN ADAMS, DAN COLLINS, KATHERINE HAWKINS, CHARMAINE SIA, ROB SILVERSMITH, AND BENA TSHISHIKU Abstract. The stick index of a knot is the least number of line segments required to build the knot in space. We define two analogous 2-dimensional invariants, the planar stick index, which is the least number of line segments in the plane to build a projection, and the spherical stick index, which is the least number of great circle arcs to build a projection on the sphere. We find bounds on these quantities in terms of other knot invariants, and give planar stick and spherical stick constructions for torus knots and for compositions of trefoils. In particular, unlike most knot invariants,we show that the spherical stick index distinguishes between the granny and square knots, and that composing a nontrivial knot with a second nontrivial knot need not increase its spherical stick index. 1. Introduction The stick index s[K] of a knot type [K] is the smallest number of straight line segments required to create a polygonal conformation of [K] in space. The stick index is generally difficult to compute. However, stick indices of small crossing knots are known, and stick indices for certain infinite categories of knots have been determined: Theorem 1.1 ([Jin97]). If Tp;q is a (p; q)-torus knot with p < q < 2p, s[Tp;q] = 2q. Theorem 1.2 ([ABGW97]). If nT is a composition of n trefoils, s[nT ] = 2n + 4. Despite the interest in stick index, two-dimensional analogues have not been studied in depth.
    [Show full text]
  • Bounds for Minimal Step Number of Knots in the Simple Cubic Lattice
    Bounds for minimal step number of knots in the simple cubic lattice R. Schareiny, K. Ishiharaz, J. Arsuagay, Y. Diao¤, K. Shimokawaz and M. Vazquezyx yDepartment of Mathematics San Francisco State University 1600 Holloway Ave San Francisco, CA 94132, USA. zDepartment of Mathematics Saitama University Saitama, 338-8570, Japan. ¤Department of Mathematics and Statistics University of North Carolina Charlotte Charlotte, NC 28223, USA. Abstract. Knots are found in DNA as well as in proteins, and they have been shown to be good tools for structural analysis of these molecules. An important parameter to consider in the arti¯cial construction of these molecules is the minimal number of monomers needed to make a knot. Here we address this problem by characterizing, both analytically and numerically, the minimum length (also called minimum step number) needed to form a particular knot in the simple cubic lattice. Our analytical work is based on an improvement of a method introduced by Diao to enumerate conformations of a given knot type for a ¯xed length. This method allows to extend the previously known result on the minimum step number of the trefoil knot 31 (which is 24) to the knots 41 and 51 and show that the minimum step numbers for the 41 and 51 knots are 30 and 34 respectively. We report on numerical results resulting from a computer implementation of this method. We provide a complete list of estimates of the minimum step numbers for prime knots up to 10 crossings, which are improvements over current published numerical results. We enumerate all minimal lattice knots of a given type and partition them into classes de¯ned by BFACF type 0 moves.
    [Show full text]
  • Upper Bounds for Equilateral Stick Numbers
    Contemporary Mathematics Upper Bounds for Equilateral Stick Numbers Eric J. Rawdon and Robert G. Scharein Abstract. We use algorithms in the software KnotPlot to compute upper bounds for the equilateral stick numbers of all prime knots through 10 cross- ings, i.e. the least number of equal length line segments it takes to construct a conformation of each knot type. We find seven knots for which we cannot construct an equilateral conformation with the same number of edges as a minimal non-equilateral conformation, notably the 819 knot. 1. Introduction Knotting and tangling appear in many physical systems in the natural sciences, e.g. in the replication of DNA. The structures in which such entanglement occurs are typically modeled by polygons, that is finitely many vertices connected by straight line segments. Topologically, the theory of knots using polygons is the same as the theory using smooth curves. However, recent research suggests that a degree of rigidity due to geometric constraints can affect the theory substantially. One could model DNA as a polygon with constraints on the edge lengths, the vertex angles, etc.. It is important to determine the degree to which geometric rigidity affects the knotting of polygons. In this paper, we explore some differences in knotting between non-equilateral and equilateral polygons with few edges. One elementary knot invariant, the stick number, denoted here by stick(K), is the minimal number of edges it takes to construct a knot equivalent to K.Richard Randell first explored the stick number in [Ran88b, Ran88a, Ran94a, Ran94b]. He showed that any knot consisting of 5 or fewer sticks must be unknotted and that stick(trefoil) = 6 and stick(figure-8) = 7.
    [Show full text]
  • Stick Number of Spatial Graphs
    STICK NUMBER OF SPATIAL GRAPHS MINJUNG LEE, SUNGJONG NO, AND SEUNGSANG OH Abstract. For a nontrivial knot K, Negami found an upper bound on the stick number s(K) in terms of its crossing number c(K) which is s(K) ≤ 2c(K). Later, Huh and Oh utilized the arc index α(K) to 3 3 present a more precise upper bound s(K) ≤ 2 c(K)+ 2 . Furthermore, Kim, No and Oh found an upper bound on the equilateral stick number s=(K) as follows; s=(K) ≤ 2c(K) + 2. As a sequel to this research program, we similarly define the stick number s(G) and the equilateral stick number s=(G) of a spatial graph G, and present their upper bounds as follows; 3 3b v s(G) ≤ c(G) + 2e + − , 2 2 2 s=(G) ≤ 2c(G) + 2e + 2b − k, where e and v are the number of edges and vertices of G, respectively, b is the number of bouquet cut-components, and k is the number of non-splittable components. 1. Introduction Throughout this paper we work in the piecewise linear category. A graph is a finite set of vertices connected by edges allowing loops and multiple edges. A spatial graph is a graph embedded in R3. We consider two spatial graphs to be the same if they are equivalent under ambient isotopy. A bouquet is a spatial graph consisting of only one vertex and loops. Note that a knot is a spatial graph consisting of a vertex and a loop. A stick spatial graph is a spatial graph which consists of finite line seg- ments, called sticks, as drawn in Figure 1.
    [Show full text]
  • The Knot Theory Course at the Ium
    THE KNOT THEORY COURSE AT THE IUM Mathematical knot theory studies certain types of curves in Euclidean space R3, namely knots, links, and braids. A knot, roughly speaking, is an oriented closed curve (without self-intersections) in Euclidean space R3 considered up to a natural equivalence relation, called isotopy. Intuitively, you should think of a knot as a thin elastic string that can be stretched, compressed, and moved about in space but may not be cut and glued back. Here are some examples of knots : (a) the unknot or trivial knot; (b) the trefoil; (c) the figure eight knot; (d) the granny knot; (e) the knot known as 52; (f) a knot isotopic to one of the first five (try to guess – which one?). Рис. 1. Examples of knots 1 2 A link, roughly speaking, is a set of several pairwise nonintersecting closed curves in R3 without self-intersections. Intuitively, you should think of a link as several thin elastic strings that can be stretched, compressed, and moved about in space but may not be cut and glued back. Here are some examples of links: (a) the trivial two component link; (b) the Hopf link; (c) the Whitehead link; (d) the Borromeo rings. Рис. 2. Examples of links Roughly speaking, a braid in n strings is an ordered set of pairwise non- intersecting curves moving downward from n aligned points of a horizontal plane to n similarly aligned points of a second horizontal plane. You can think of the strings of a braid as being thin elastic strings can be stretched, compressed, and moved about in space, but may not be cut and glued back.
    [Show full text]
  • New Stick Number Bounds from Random Sampling of Confined Polygons
    New Stick Number Bounds from Random Sampling of Confined Polygons Thomas D. Eddy and Clayton Shonkwiler Department of Mathematics, Colorado State University, Fort Collins, CO Abstract The stick number of a knot is the minimum number of segments needed to build a polygonal version of the knot. Despite its elementary definition and relevance to physical knots, the stick number is poorly understood: for most knots we only know bounds on the stick number. We adopt a Monte Carlo approach to finding better bounds, producing very large ensembles of random polygons in tight confinement to look for new examples of knots constructed from few segments. We generated a total of 220 billion random polygons, yielding either the exact stick number or an improved upper bound for more than 40% of the knots with 10 or fewer crossings for which the stick number was not previously known. We summarize the current state of the art in Appendix A, which gives the best known bounds on stick number for all knots up to 10 crossings. 1 Introduction The stick number of a knot is the minimum number of segments needed to create a polygonal version of the knot. The equilateral stick number is defined similarly, though with the added restriction that all the segments should be the same length. Despite their elementary definitions, these invariants are only poorly understood: for example, prior to our work the exact stick number was only known for 30 of the 249 nontrivial knots up to 10 crossings. The sheer simplicity of its definition partially explains the appeal of the stick number as a knot invariant, but the stick number is also interesting as a prototypical geometric or physical knot invariant since the number of segments needed to build a knot is a measure of the geometric complexity of the knot.
    [Show full text]
  • Arxiv:2010.04188V1 [Math.GT] 8 Oct 2020 Knot Has Been “Pulled Tight”, Minimizing the Folded Ribbonlength
    RIBBONLENGTH OF FAMILIES OF FOLDED RIBBON KNOTS ELIZABETH DENNE, JOHN CARR HADEN, TROY LARSEN, AND EMILY MEEHAN ABSTRACT. We study Kauffman’s model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a ribbon knot. We give upper bounds on the folded ribbonlength of 2-bridge, (2; p) torus, twist, and pretzel knots, and these upper bounds turn out to be linear in crossing number. We give a new way to fold (p; q) torus knots, and show that their folded ribbonlength is bounded above by p + q. This means, for example, that the trefoil knot can be constructed with a folded ribbonlength of 5. We then show that any (p; q) torus knot K has a constant c > 0, such that the folded ribbonlength is bounded above by c · Cr(K)1=2, providing an example of an upper bound on folded ribbonlength that is sub-linear in crossing number. 1. INTRODUCTION Take a long thin strip of paper, tie a trefoil knot in it then gently tighten and flatten it. As can be seen in Figure 1, the boundary of the knot is in the shape of a pentagon. This observation is well known in recreational mathematics [4, 21, 34]. L. Kauffman [27] introduced a mathematical model of such a folded ribbon knot. Kauffman viewed the ribbon as a set of rays parallel to a polygo- nal knot diagram with the folds acting as mirrors, and the over-under information appropriately preserved.
    [Show full text]
  • The Knot Spectrum of Random Knot Spaces
    Yuanan Diao, Claus Ernst*, Uta Ziegler, and Eric J. Rawdon The Knot Spectrum of Random Knot Spaces Abstract: It is well known that knots exist in natural systems. For example, in the case of (mutant) bacteriophage P4, DNA molecules packed inside the bacteriophage head are considered to be circular since the two sticky ends of the DNA are close to each other. The DNAs extracted from the capsid, without separating the two ends, can pre- serve the topology of the (circular) DNAs, and hence are well-defined knots. Further- more, knots formed within such systems are often varied and different knots occur with different probabilities. Such information can be important in biology. Mathemat- ically, we may view (and model) such a biological system as (by) a random knot space and attempt to obtain information about the system via mathematical analysis and numerical simulation. The question here is to find the probability that a randomly (and uniformly) chosen knot from this space is of a particular knot type. This is equiv- alent to finding the distribution of all knot types within this random knot space (called the knot spectrum in an earlier paper by the authors). In this paper, we examine the behavior of the knot spectrums for knots up to 10 crossings. Using random polygons of various lengths under different confinement conditions as the random knot spaces (model biological systems), we demonstrate that the relative spectrums of the knots, when divided into groups by their crossing numbers, remain surprisingly robust as these knot spaces vary. For a given knot type K, we let PK(L, R) be the probability that an equilateral random polygon of length L in a confinement sphere of radius R has knot type K.
    [Show full text]
  • Arxiv:1512.03592V1 [Math.GT] 11 Dec 2015 Httease O H Rtqeto Sngtv O H Knot the for Negative Is [5]
    AN UPPER BOUND ON STICK NUMBERS OF KNOTS YOUNGSIK HUH AND SEUNGSANG OH Abstract. In 1991, Negami found an upper bound on the stick number s(K) of a nontrivial knot K in terms of the minimal crossing number c(K) of the knot which is s(K) ≤ 2c(K). s K 3 c K K In this paper we improve this upper bound to ( ) ≤ 2 ( ( ) + 1). Moreover if is a s K 3 c K non-alternating prime knot, then ( ) ≤ 2 ( ). 1. Introduction A simple closed curve embedded into the Euclidean 3-space is called a knot. Two knots K and K′ are said to be equivalent, if there exists an orientation preserving homeomorphism of R3 which maps K to K′, or to say roughly, we can obtain K′ from K by a sequence of moves without intersecting any strand of the knot. And the equivalence class of K is called the knot type of K. A knot equivalent to another knot in a plane of the 3-space is said to be trivial. A stick knot is a knot which consists of finite line segments, called sticks. One natural question concerning stick knots may be the stick number s(K) of a knot K which is defined to be the minimal number of sticks for construction of the knot type into a stick knot. Since this representation of knots has been considered to be a useful mathematical model of cyclic molecules or molecular chains, the stick number may be an interesting quantity not only in knot theory of mathematics, but also in chemistry and physics.
    [Show full text]
  • Properties of Glued Knots
    Properties of glued knots Shane D'Mello1 and Vinay Gaba1 1Indian Institute of Science Education and Research, Mohali, India July 28, 2021 Abstract We study the properties of, glued knots, a sub-class of real rational knots that can be constructed by gluing ellipses. We define an invariant called the gluing degree and relate it to various classical properties of knots and classify all knots up to gluing degree 6. 1 Introduction The study of real algebraic knots is one of the tangential questions motivated by Hilbert's Sixteenth problem. Until now, little is known about the classical knot theoretic properties of such knots and the question seems hard to answer in general. In [1], Bj¨orklundintroduced the method of constructing a real rational knot of degree d1 + d2 by gluing two real rational knots of degrees d1 and d2. We recall the gluing construction: Theorem 1.1 (Bj¨orklund). Consider two oriented real rational knots k1 and k2 of degree d1 and d2 3 respectively in RP which intersect transversally in a point p. Therefore, in a small enough neighbourhood of p the union is isotopic to a pair of intersecting straight line segments. Consider the knot obtained by smoothing the pair of segments in a manner compatible with the orientation, then the resulting knot has a real rational representative of degree d1 + d2 which lies completely inside a small tubular neighbourhood of the union of k1 and k2. In [2], it was shown that all classical knots have an isotopy class representative that can be constructed using this method and one can know the degree of the resulting real rational knot that is sufficient.
    [Show full text]