DOCSLIB.ORG

The Hopf Fibration and Encoding Torus Knots in Light Fields

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Hopf Fibration and Encoding Torus Knots in Light Fields UNLV Theses, Dissertations, Professional Papers, and Capstones May 2016 The Hopf Fibration and Encoding Torus Knots in Light Fields John Vincent Waite University of Nevada, Las Vegas Follow this and additional works at: https://digitalscholarship.unlv.edu/thesesdissertations Part of the Physics Commons Repository Citation Waite, John Vincent, "The Hopf Fibration and Encoding Torus Knots in Light Fields" (2016). UNLV Theses, Dissertations, Professional Papers, and Capstones. 2756. http://dx.doi.org/10.34917/9112204 This Thesis is protected by copyright and/or related rights. It has been brought to you by Digital Scholarship@UNLV with permission from the rights-holder(s). You are free to use this Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Thesis has been accepted for inclusion in UNLV Theses, Dissertations, Professional Papers, and Capstones by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. THE HOPF FIBRATION AND ENCODING TORUS KNOTS IN LIGHT FIELDS By John Waite Bachelor of Science - Mathematics University of Maryland, College Park 2010 A thesis submitted in partial fulfillment of the requirements for the Master of Science - Physics Department of Physics and Astronomy College of Sciences The Graduate College University of Nevada, Las Vegas May 2016 Thesis Approval The Graduate College The University of Nevada, Las Vegas April 8, 2016 This thesis prepared by John Waite entitled The Hopf Fibration and Encoding Torus Knots in Light Fields is approved in partial fulfillment of the requirements for the degree of Master of Science – Physics Department of Physics and Astronomy Bernard Zygelman, Ph.D. Kathryn Hausbeck Korgan, Ph.D. Examination Committee Chair Graduate College Interim Dean Michael Pravica, Ph.D. Examination Committee Member Stephen Lepp, Ph.D. Examination Committee Member Dieudonne D. Phanord, Ph.D. Graduate College Faculty Representative ii Abstract The first sections of this thesis explore a construction of electromagnetic fields first proposed by Bateman [2] and recently rediscovered [8] and expanded upon. With this powerful and elegant tool, I show that it is possible to construct families of EM fields that have a common topological structure that is preserved throughout time. This topological structure, known as the Hopf fibration, has been found to manifest itself in many areas of physics. Due to its utility, I have made a detailed study of it. The final section of this thesis develops an algorithm to parameterize a closed, bounded, and oriented surface that has as its boundary atorusknot.Thesetypesofsurfaces,knownasSeifertsurfaces[17],containinformation about the energy structure of the EM fields developed in the earlier sections of this thesis and may also lead to detection protocols for such fields. Because these surfaces are closed, bounded and oriented then Stokes’ theorem should be valid. A verification of this proposition is included. iii Acknowledgements I would like to thank my advisor, Dr. Bernard Zygelman, for the guidance and direction he provided me throughout the entire process of producing this thesis, and also for supplying me with the ideas in the first place. I would also like to acknowledge the other members of my committee, Drs. Lepp, Pravica, and Phanord, for their patience with me during the weeks leading to my defense. Finally, I would like to thank my wife, Kelly, for the sacrifices she was required to make in my behalf for the past two years. iv Dedication This thesis is dedicated to my dear wife Kelly. Without your support this project would never have started in the first place. v Table of Contents Abstract iii Acknowledgements iv Dedication v Table of Contents vi List of Figures viii 1 Introduction 1 2 Bateman’s Construction 3 2.1 Motivation . 3 2.2 Derivation . 3 2.3 Remarks . 9 3 Elliptically Polarized Plane Wave 10 3.1 Parameters . 10 3.2 Verifications . 11 3.3 Remarks . 12 4 The Torus 13 5 The Hopfion Solution 15 5.1 Introduction . 15 5.2 TheHopfFibration................................ 16 5.3 Parameters . 20 5.4 Verifications . 27 5.5 Remarks . 28 vi 6 Generalized Solutions on Torus Knots 29 6.1 Parameters . 29 6.2 Remarks . 31 7 Seifert Surfaces 32 7.1 ‘Top’and‘Bottom’Surfaces . 33 7.2 Twisted Bands . 37 7.3 Verification of Stokes’ Theorem . 40 8 Conclusions 45 Appendix A Parameterizing a Seifert Surface of the Trefoil Knot 46 Appendix B Verification of Stokes’ Theorem 50 Appendix C Visualizations of the Hopf Fibration 56 References 59 Curriculum Vitae 61 vii List of Figures 1The2-torusandtheplanearetopologicallyequivalent.............13 2 Nestedtori..................................... 14 3TheRiemannSphere...............................19 4 ThefibersoftheHopfmaparelinkedcircles . 21 5Latitudesproducetori..............................21 6 TheHopffibration ................................ 22 7ThetorusT 2 with component circles shown . 33 8Deletingcrossings.................................33 9 Twodiscsinparallelplanes ........................... 34 10 Generalization of the top and bottom surfaces . 35 11 Parameterizing a disc . 36 12 Actualtopandbottomsurfaces . 37 13 The boundary of the band . 38 14 Twisted bands . 39 15 The finished Seifert surface for the trefoil knot . 39 16 Line elements for the contour integral . 41 17 Vectorsnormaltoasurface ........................... 42 18 Full surface with field . 44 viii 1 Introduction The study of knots has a rich history, dating back to the 1860’s when Lord Kelvin proposed his vortex atom hypothesis [18]. Although this hypothesis was eventually abandoned with the discovery of the atom about a decade later, the rigorous study of knots in a mathematical framework was well underway. Until recently, the study of knots was confined almost exclu- sively to the realm of pure mathematics. Knots, and their intrinsic topological properties, have again entered the domain of mainstream theoretical physics. The remarkable aspect of the solutions of Maxwell’s equations we will investigate are their stablity in time, and that they possess a topological invariant, in that the electric and magnetic field lines are linked for all time, and the Poynting field line structure is unchanged in time. This topological structure could be exploited in many applications. The study of topological invariants has recently gained ground in many branches of physics research. Notably, the advances made in the field of quantum computation and information have yielded promising results that utilize the topological nature of light and matter [4,10]. While much of this research is still confined to the realm of theoretical physics, it will not be long before these phenomenon may be realized in the laboratory. This recent activity has resulted in a large increase in the funding available for continued research and development. The solutions studied in this paper are confined to the realm of electromagnetic radiation, but it would not be prudent to limit the study of such solutions to light. There is no reason to assume that these types of solutions could not be applied to matter as well. The reason that such topological invariants are important to such branches of study such as quantum information lies in the stability and robustness of that topology. These types of invariants are nearly immune to noise and perturbations, making them ideal candidates for information storage and use as qubits, as they are essentially deaf to sources of decoherence. In Chapter 2, I will present the construction proposed by Bateman [2]. Chapter 3 will verify the validity of this construction on a familiar solution to Maxwell’s equations in free space. Chapters 4 and 5 provide some background on tori and a presentation of the structure 1 referred to as the Hopf fibration. Chapter 6 will examine the solutions proposed by Kedia et al. [8]. Finally, Chapter 7 will present an algorithm to parameterize Seifert surfaces and verify Stokes’ theorem on one such surface. 2 2 Bateman’s Construction 2.1 Motivation The usual solutions to Maxwell’s equations we encounter are plane waves. These solutions, while important and useful, are only a small subset of the total solutions that are allowed. Finding and developing solutions to the wave equation has been a part of physics education and continuing research for the past 150 years. The reason for this restricted subset of solu- tions arises from the methods used to solve the resulting wave equation. It is typical to use the method of separation of variables to find solutions. However, there is no fundamental reason for all solutions to be separable. With the advances made in technology, specifically in computation, we have been able to shed light on and further our understanding of these non-separable solutions. One of the basic principles that makes this field a nearly inex- haustible one is the fact that the superposition of solutions is also a solution. This principle of superposition, and accounting for various boundary conditions, is vital when we attempt to create new solutions from old ones. The following derivation explores another method to solving the wave equation. 2.2 Derivation The method outlined in the thesis uses the construction of an electromagnetic field proposed by Bateman [2] as early as 1915. This construction goes as follows.
Load more

Recommended publications
  • Maps That Take Lines to Circles, in Dimension 4
    Maps That Take Lines to Circles, in Dimension 4 V. Timorin Abstract. We list all analytic diffeomorphisms between an open subset of the 4-dimen- sional projective space and an open subset of the 4-dimensional sphere that take all line segments to arcs of round circles. These are the following: restrictions of the quaternionic Hopf fibrations and projections from a hyperplane to a sphere from some point. We prove this by finding the exact solutions of the corresponding system of partial differential equations. 1 Introduction Let U be an open subset of the 4-dimensional real projective space RP4 and V an open subset of the 4-dimensional sphere S4. We study diffeomorphisms f : U → V that take all line segments lying in U to arcs of round circles lying in V . For the sake of brevity we will always say in the sequel that f takes all lines to circles. The purpose of this article is to give the complete list of such analytic diffeomorphisms. Remark. Given a diffeomorphism f : U → V that takes lines to circles, we can compose it with a projective transformation in the preimage (which takes lines to lines) and a conformal transformation in the image (which takes circles to circles). The result will be another diffeomorphism taking lines to circles. Example 1. For example, suppose that S4 is embedded in R5 as a Euclidean sphere and take an arbitrary hyperplane and an arbitrary point in R5. Obviously, the pro- jection of the hyperplane to S4 form the given point takes all lines to circles.
    [Show full text]
  • An Introduction to Knot Theory and the Knot Group
    AN INTRODUCTION TO KNOT THEORY AND THE KNOT GROUP LARSEN LINOV Abstract. This paper for the University of Chicago Math REU is an expos- itory introduction to knot theory. In the first section, definitions are given for knots and for fundamental concepts and examples in knot theory, and motivation is given for the second section. The second section applies the fun- damental group from algebraic topology to knots as a means to approach the basic problem of knot theory, and several important examples are given as well as a general method of computation for knot diagrams. This paper assumes knowledge in basic algebraic and general topology as well as group theory. Contents 1. Knots and Links 1 1.1. Examples of Knots 2 1.2. Links 3 1.3. Knot Invariants 4 2. Knot Groups and the Wirtinger Presentation 5 2.1. Preliminary Examples 5 2.2. The Wirtinger Presentation 6 2.3. Knot Groups for Torus Knots 9 Acknowledgements 10 References 10 1. Knots and Links We open with a definition: Definition 1.1. A knot is an embedding of the circle S1 in R3. The intuitive meaning behind a knot can be directly discerned from its name, as can the motivation for the concept. A mathematical knot is just like a knot of string in the real world, except that it has no thickness, is fixed in space, and most importantly forms a closed loop, without any loose ends. For mathematical con- venience, R3 in the definition is often replaced with its one-point compactification S3. Of course, knots in the real world are not fixed in space, and there is no interesting difference between, say, two knots that differ only by a translation.
    [Show full text]
  • Coefficients of Homfly Polynomial and Kauffman Polynomial Are Not Finite Type Invariants
    COEFFICIENTS OF HOMFLY POLYNOMIAL AND KAUFFMAN POLYNOMIAL ARE NOT FINITE TYPE INVARIANTS GYO TAEK JIN AND JUNG HOON LEE Abstract. We show that the integer-valued knot invariants appearing as the nontrivial coe±cients of the HOMFLY polynomial, the Kau®man polynomial and the Q-polynomial are not of ¯nite type. 1. Introduction A numerical knot invariant V can be extended to have values on singular knots via the recurrence relation V (K£) = V (K+) ¡ V (K¡) where K£, K+ and K¡ are singular knots which are identical outside a small ball in which they di®er as shown in Figure 1. V is said to be of ¯nite type or a ¯nite type invariant if there is an integer m such that V vanishes for all singular knots with more than m singular double points. If m is the smallest such integer, V is said to be an invariant of order m. q - - - ¡@- @- ¡- K£ K+ K¡ Figure 1 As the following proposition states, every nontrivial coe±cient of the Alexander- Conway polynomial is a ¯nite type invariant [1, 6]. Theorem 1 (Bar-Natan). Let K be a knot and let 2 4 2m rK (z) = 1 + a2(K)z + a4(K)z + ¢ ¢ ¢ + a2m(K)z + ¢ ¢ ¢ be the Alexander-Conway polynomial of K. Then a2m is a ¯nite type invariant of order 2m for any positive integer m. The coe±cients of the Taylor expansion of any quantum polynomial invariant of knots after a suitable change of variable are all ¯nite type invariants [2]. For the Jones polynomial we have Date: October 17, 2000 (561).
    [Show full text]
  • The Hopf Fibration
    THE HOPF FIBRATION The Hopf fibration is an important object in fields of mathematics such as topology and Lie groups and has many physical applications such as rigid body mechanics and magnetic monopoles. This project will introduce the Hopf fibration from the points of view of the quaternions and of the complex numbers. n n+1 Consider the standard unit sphere S ⊂ R to be the set of points (x0; x1; : : : ; xn) that satisfy the equation 2 2 2 x0 + x1 + ··· + xn = 1: One way to define the Hopf fibration is via the mapping h : S3 ! S2 given by (1) h(a; b; c; d) = (a2 + b2 − c2 − d2; 2(ad + bc); 2(bd − ac)): You should check that this is indeed a map from S3 to S2. 3 (1) First, we will use the quaternions to study rotations in R . As a set and as a 4 vector space, the set of quaternions is identical to R . There are 3 distinguished coordinate vectors{(0; 1; 0; 0); (0; 0; 1; 0); (0; 0; 0; 1){which are given the names i; j; k respectively. We write the vector (a; b; c; d) as a + bi + cj + dk. The multiplication rules for quaternions can be summarized via the following: i2 = j2 = k2 = −1; ij = k; jk = i; ki = j: Is quaternion multiplication commutative? Is it associative? We can define several other notions associated with quaternions. The conjugate of a quaternionp r = a + bi + cj + dk isr ¯ = a − bi − cj − dk. The norm of r is jjrjj = a2 + b2 + c2 + d2.
    [Show full text]
  • CALIFORNIA STATE UNIVERSITY, NORTHRIDGE P-Coloring Of
    CALIFORNIA STATE UNIVERSITY, NORTHRIDGE P-Coloring of Pretzel Knots A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics By Robert Ostrander December 2013 The thesis of Robert Ostrander is approved: |||||||||||||||||| |||||||| Dr. Alberto Candel Date |||||||||||||||||| |||||||| Dr. Terry Fuller Date |||||||||||||||||| |||||||| Dr. Magnhild Lien, Chair Date California State University, Northridge ii Dedications I dedicate this thesis to my family and friends for all the help and support they have given me. iii Acknowledgments iv Table of Contents Signature Page ii Dedications iii Acknowledgements iv Abstract vi Introduction 1 1 Definitions and Background 2 1.1 Knots . .2 1.1.1 Composition of knots . .4 1.1.2 Links . .5 1.1.3 Torus Knots . .6 1.1.4 Reidemeister Moves . .7 2 Properties of Knots 9 2.0.5 Knot Invariants . .9 3 p-Coloring of Pretzel Knots 19 3.0.6 Pretzel Knots . 19 3.0.7 (p1, p2, p3) Pretzel Knots . 23 3.0.8 Applications of Theorem 6 . 30 3.0.9 (p1, p2, p3, p4) Pretzel Knots . 31 Appendix 49 v Abstract P coloring of Pretzel Knots by Robert Ostrander Master of Science in Mathematics In this thesis we give a brief introduction to knot theory. We define knot invariants and give examples of different types of knot invariants which can be used to distinguish knots. We look at colorability of knots and generalize this to p-colorability. We focus on 3-strand pretzel knots and apply techniques of linear algebra to prove theorems about p-colorability of these knots.
    [Show full text]
  • Alexander Polynomial, Finite Type Invariants and Volume of Hyperbolic
    ISSN 1472-2739 (on-line) 1472-2747 (printed) 1111 Algebraic & Geometric Topology Volume 4 (2004) 1111–1123 ATG Published: 25 November 2004 Alexander polynomial, finite type invariants and volume of hyperbolic knots Efstratia Kalfagianni Abstract We show that given n > 0, there exists a hyperbolic knot K with trivial Alexander polynomial, trivial finite type invariants of order ≤ n, and such that the volume of the complement of K is larger than n. This contrasts with the known statement that the volume of the comple- ment of a hyperbolic alternating knot is bounded above by a linear function of the coefficients of the Alexander polynomial of the knot. As a corollary to our main result we obtain that, for every m> 0, there exists a sequence of hyperbolic knots with trivial finite type invariants of order ≤ m but ar- bitrarily large volume. We discuss how our results fit within the framework of relations between the finite type invariants and the volume of hyperbolic knots, predicted by Kashaev’s hyperbolic volume conjecture. AMS Classification 57M25; 57M27, 57N16 Keywords Alexander polynomial, finite type invariants, hyperbolic knot, hyperbolic Dehn filling, volume. 1 Introduction k i Let c(K) denote the crossing number and let ∆K(t) := Pi=0 cit denote the Alexander polynomial of a knot K . If K is hyperbolic, let vol(S3 \ K) denote the volume of its complement. The determinant of K is the quantity det(K) := |∆K(−1)|. Thus, in general, we have k det(K) ≤ ||∆K (t)|| := X |ci|. (1) i=0 It is well know that the degree of the Alexander polynomial of an alternating knot equals twice the genus of the knot.
    [Show full text]
  • Results on Nonorientable Surfaces for Knots and 2-Knots
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Dissertations, Theses, and Student Research Papers in Mathematics Mathematics, Department of 8-2021 Results on Nonorientable Surfaces for Knots and 2-knots Vincent Longo University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/mathstudent Part of the Geometry and Topology Commons Longo, Vincent, "Results on Nonorientable Surfaces for Knots and 2-knots" (2021). Dissertations, Theses, and Student Research Papers in Mathematics. 111. https://digitalcommons.unl.edu/mathstudent/111 This Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Dissertations, Theses, and Student Research Papers in Mathematics by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. RESULTS ON NONORIENTABLE SURFACES FOR KNOTS AND 2-KNOTS by Vincent Longo A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy Major: Mathematics Under the Supervision of Professors Alex Zupan and Mark Brittenham Lincoln, Nebraska May, 2021 RESULTS ON NONORIENTABLE SURFACES FOR KNOTS AND 2-KNOTS Vincent Longo, Ph.D. University of Nebraska, 2021 Advisor: Alex Zupan and Mark Brittenham A classical knot is a smooth embedding of the circle S1 into the 3-sphere S3. We can also consider embeddings of arbitrary surfaces (possibly nonorientable) into a 4-manifold, called knotted surfaces. In this thesis, we give an introduction to some of the basics of the studies of classical knots and knotted surfaces, then present some results about nonorientable surfaces bounded by classical knots and embeddings of nonorientable knotted surfaces.
    [Show full text]
  • Unknotting Sequences for Torus Knots
    CORE Metadata, citation and similar papers at core.ac.uk Provided by RERO DOC Digital Library Math. Proc. Camb. Phil. Soc. (2010), 148, 111 c Cambridge Philosophical Society 2009 111 doi:10.1017/S0305004109990156 First published online 6 July 2009 Unknotting sequences for torus knots BY SEBASTIAN BAADER Department of Mathematics, ETH Zurich,¨ Switzerland. e-mail: [email protected] (Received 3 April 2009; revised 12 May 2009) Abstract The unknotting number of a knot is bounded from below by its slice genus. It is a well- known fact that the genera and unknotting numbers of torus knots coincide. In this paper we characterize quasipositive knots for which the genus bound is sharp: the slice genus of a quasipositive knot equals its unknotting number, if and only if the given knot appears in an unknotting sequence of a torus knot. 1. Introduction The unknotting number is a classical measure of complexity for knots. It is defined as the minimal number of crossing changes needed to transform a given knot into the trivial knot [14]. An estimate of the unknotting number is usually difficult, even for special classes of knots such as torus knots. It is the content of the Milnor conjecture, proved by Kronheimer and Mrowka [7], that the unknotting number of torus knots coincides with another classical invariant of knots, the slice genus. The slice genus g∗(K ) of a knot K is the minimal genus among all oriented surfaces smoothly embedded in the 4-ball with boundary K . In general, the slice genus of a knot K provides a lower bound for its unknotting number u(K ): u(K ) g∗(K ).
    [Show full text]
  • Hopf Fibration and Clifford Translation* of the 3-Sphere See Clifford Tori
    Hopf Fibration and Clifford Translation* of the 3-sphere See Clifford Tori and their discussion first. Most rotations of the 3-dimensional sphere S3 are quite different from what we might expect from familiarity with 2-sphere rotations. To begin with, most of them have no fixed points, and in fact, certain 1-parameter subgroups of rotations of S3 resemble translations so much, that they are referred to as Clifford translations. The description by formulas looks nicer in complex notation. For this we identify R2 with C, as usual, and multiplication by i in C 0 1 represented in 2 by matrix multiplication by . R 1 −0 µ ∂ Then the unit sphere S3 in R4 is given by: 3 2 2 2 S := p = (z1, z2) C ; z1 + z2 = 1 { ∈ | | | | } 4 2 (x1, x2, x3, x4) R ; (xk) = 1 . ∼ { ∈ } X And for ϕ R we define the Clifford Translation Cϕ : 3 3 ∈ iϕ iϕ S S by Cϕ(z1, z2) := (e z1, e z2). → The orbits of the one-parameter group Cϕ are all great circles, and they are equidistant from each other in analogy to a family of parallel lines; it is because of this behaviour that the Cϕ are called Clifford translations. * This file is from the 3D-XplorMath project. Please see: http://3D-XplorMath.org/ 29 But in another respect the behaviour of the Cϕ is quite different from a translation – so different that it is diffi- cult to imagine in R3. At each point p S3 we have one ∈ 2-dimensional subspace of the tangent space of S3 which is orthogonal to the great circle orbit through p.
    [Show full text]
  • Determinants of Amphichiral Knots
    DETERMINANTS OF AMPHICHIRAL KNOTS STEFAN FRIEDL, ALLISON N. MILLER, AND MARK POWELL Abstract. We give a simple obstruction for a knot to be amphichiral, in terms of its determinant. We work with unoriented knots, and so obstruct both positive and negative amphichirality. 1. Introduction By a knot we mean a 1-dimensional submanifold of S3 that is diffeomorphic to S1. Given a knot K we denote its mirror image by mK, the image of K under an orientation reversing homeomorphism S3 ! S3. We say that a knot K is amphichiral if K is (smoothly) isotopic to mK. Note that we consider unoriented knots, so we do not distinguish between positive and negative amphichiral knots. Arguably the simplest invariant of a knot K is the determinant det(K) 2 Z which can be introduced in many different ways. The definition that we will work with is that det(K) is the order of the first homology of the 2-fold cover Σ(K) of S3 branched along K. Alternative definitions are given by det(K) = ∆K (−1) = JK (−1) where ∆K (t) denotes the Alexander polynomial and JK (q) denotes the Jones polynomial [Li97, Corollary 9.2], [Ka96, Theorem 8.4.2]. We also recall that for a knot K with Seifert matrix A, a presentation T matrix for H1(Σ(K)) is given by A + A . In particular one can readily compute the determinant via det(K) = det(A + AT ). The following proposition gives an elementary obstruction for a knot to be amphichiral. Proposition 1.1. Suppose K is an amphichiral knot and p is a prime with p ≡ 3 mod 4.
    [Show full text]
  • Jones Polynomial of Knots
    KNOTS AND THE JONES POLYNOMIAL MATH 180, SPRING 2020 Your task as a group, is to research the topics and questions below, write up clear notes as a group explaining these topics and the answers to the questions, and then make a video presenting your findings. Your video and notes will be presented to the class to teach them your findings. Make sure that in your notes and video you give examples and intuition, along with formal definitions, theorems, proofs, or calculations. Make sure that you point out what the is most important take away message, and what aspects may be tricky or confusing to understand at first. You will need to work together as a group. You should all work on Problem 1. Each member of the group must be responsible for one full example from problem 2. Then you can split up problem 3-7 as you wish. 1. Resources The primary resource for this project is The Knot Book by Colin Adams, Chapter 6.1 (page 147-155). An Introduction to Knot Theory by Raymond Lickorish Chapter 3, could also be helpful. You may also look at other resources online about knot theory and the Jones polynomial. Make sure to cite the sources you use. If you find it useful and you are comfortable, you can try to write some code to help you with computations. 2. Topics and Questions As you research, you may find more examples, definitions, and questions, which you defi- nitely should feel free to include in your notes and/or video, but make sure you at least go through the following discussion and questions.
    [Show full text]
  • Geometric Spinors, Relativity and the Hopf Fibration
    Geometric Spinors, Relativity and the Hopf Fibration Garret Sobczyk Universidad de las Americas-Puebla´ Departamento de F´ısico-Matematicas´ 72820 Puebla, Pue., Mexico´ http://www.garretstar.com September 26, 2015 Abstract This article explores geometric number systems that are obtained by extending the real number system to include new anticommuting square roots of ±1, each such new square root representing the direction of a unit vector along orthogonal coordinate axes of a Euclidean or pesudoEuclidean space. These new number systems can be thought of as being nothing more than a geometric basis for tables of numbers, called matrices. At the same time, the consistency of matrix algebras prove the consistency of our geometric number systems. The flexibility of this new concept of geometric numbers opens the door to new understanding of the nature of spacetime, the concept of Pauli and Dirac spinors, and the famous Hopf fibration. AMS Subject Classification: 15A66, 81P16 Keywords: geometric algebra, spacetime algebra, Riemann sphere, relativity, spinor, Hopf fibration. 1 Introduction The concept of number has played a decisive role in the ebb and flow of civilizations across centuries. Each more advanced civilization has made its singular contributions to the further development, starting with the natural “counting numbers” of ancient peoples, to the quest of the Pythagoreans’ idea that (rational) numbers are everything, to the heroic development of the “imaginary” numbers to gain insight into the solution of cubic and quartic polynomials, which underlies much of modern mathematics, used extensively by engineers, physicists and mathematicians of today [4]. I maintain that the culmination of this development is the geometrization of the number concept: Axiom: The real number system can be geometrically extended to include new, anti-commutative square roots of ±1, each new such square root rep- resenting the direction of a unit vector along orthogonal coordinate axes 1 of a Euclidean or pseudo-Euclidean space Rp;q.
    [Show full text]