DOCSLIB.ORG

Knot Polynomial Invariants in Terms of Helicity for Tackling Topology of Fluid

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Motivations Knot polynomials in terms of helicity Examples Outlook Topological Dynamics in the Physical and Biological Sciences Knot polynomial invariants in terms of helicity for tackling topology of fluid knots Xin LIU School of Mathematics and Statistics University of Sydney, NSW 2006, Australia [email protected] Joint work with: Renzo L. Ricca (University of Milano-Bicocca, Italy) 1 November 2012, Isaac Newton Institute, Cambridge Motivations Knot polynomials in terms of helicity Examples Outlook CONTENTS • Motivations • Determining topology of fluid knots • Heuristics of quantum Chern-Simons theory • Exponential form of helicity • Knot polynomial invariants in terms of helicity • Main Theorem • Revisit to knot polynomials and realization of skein relations • Proof of theorem • Examples: computing the Jones polynomials of typical knots/links • Trefoil knots • Figure-of-eight knot • Whitehead link • Borromean rings • Outlook Motivations Knot polynomials in terms of helicity Examples Outlook Motivations • Determination of fluid knot topology needs powerful tools investigations on =⇒ knot polynomial invariants • Heuristics of quantum Chern-Simons eHelicity theory Motivations Knot polynomials in terms of helicity Examples Outlook Topology is important: how tangled is a fluid knot? TRACE (NASA) Uddin et al. (2009) Magnetic fields and vorticity localization occur in classical and quantum fluids. Topology of fluid knots is crucial, as it is related to energy and dynamical properties. How to characterise topology of fluid knots? Barenghi et al. (2001) Raymer & Smith (2007) Motivations Knot polynomials in terms of helicity Examples Outlook Current approach: linking numbers • Helicity of vortex flows (Woltjer, 1958) Z H = u · ω d3x, u — velocity, ω — vorticity Ω 3 3 ω = ∇ × u on the support Ω ⊂ R ; ∇ · u = 0 in R ; ω · nˆ = 0 on ∂Ω. • For a link K of N vortex knots of circulation κi, we have (Moffatt 1969; Ricca & Moffatt 1992) X 2 X H (K) = κi Sli + 2 κiκj Lkij i ij Sli —Calug˘ areanu-White˘ self-linking number Lkij — Gauss linking number • However, sometimes linking numbers fail to detect essential topology: Motivations Knot polynomials in terms of helicity Examples Outlook Why helicity? To solve this problem, our idea is to focus on helicity: • Helicity is the most important topological invariant in fluid mechanics. Thanks to this Newton Institute program (— IUTAM & Edinburgh workshops): • Helicity plays an central role in the research of fluid mechanics: theoretical and experimental • Higher order linking numbers and Kontsevich integrals: • Cohomological class: Generalized helicity based on generalized Gauss mapping • Helicity has remarkable geometric significance — Abelian Chern-Simons 3-form (or abelian Chern-Simons action) Motivations Knot polynomials in terms of helicity Examples Outlook Chern-Simons theory • Generic Chern-Simons action with a non-abelian gauge group G: k Z 2 S = A ∧ dA + A ∧ A ∧ A, M 3— 3-manifold. 4π M 3 3 a where A is the gauge potential, A = A Ta, with Ta denoting the generators of G. The cubic term A ∧ A ∧ A labels the non-abelianity. • Fluid mechanics is treated as an abelian case, namely, the gauge group is U(1). • Helicity, as an isotopy invariant, is defined as Z H = u · ω d3x, ω = ∇ × u. Ω In the language of differential forms and wedge products: i u =⇒ 1-form u = uidx , ω =⇒ 2-form ω, ω = du. The helicity becomes a 3-form integral: Z H = u ∧ du. Ω • Hence, helicity is an abelian Chern-Simons action. Motivations Knot polynomials in terms of helicity Examples Outlook Extraordinary success of quantum Chern-Simons theory • The past two decades have witnessed the remarkable progress of quantum CS theory as the most important 3-dimensional topological quantum field theory. It provides a field theoretical justification for knot and 3-manifold topological invariants beyond algebraic quantum groups. • Witten’s seminal work (1989; Fields Medal 1990) =⇒ Two major research approaches in 3D TQFT: • algebra-geometric description for conformal field theory; • Reshetikhin-Turaev constructions from tensor category. • The third major approach in 3D TQFT is perturbative expansions of CS integrals. • Bar-Natan, Birman-Lin, Guadagnini, Labastida, Le-Murakami-Ohtsuki, et al. • Exponential expansions =⇒ Vassiliev invariants, via the Kontsevich integral. • Birman & Lin (1993): The Jones, HOMFLY and Kauffman polynomials exponential form −→ Vassiliev invariants (finite-type invariants) Motivations Knot polynomials in terms of helicity Examples Outlook Powerful tools of knot theory: knot polynomial invariants • Knot polynomials: Alexander-Conway, Jones, Kauffman, HOMFLY, . • Vassiliev finite type invariants, . Why no such tools in fluid mechanics? — complex flux tube structures (bundle of vortex lines)? — thousands of crossing sites? Motivations Knot polynomials in terms of helicity Examples Outlook Heuristics • In quantum Chern-Simons theory: Topological invariants ⇐= Vacuum expectation value of Wilson loops * N H + Z P H Y i A 1 i A iS e γk = [DA] e k γk e Z — partition function Z k=1 • In the case of superfluid, helicity turns to be line integrals Z X I H (K) = u ∧ du = κi u · dl, Ω i γi K = {γi}, γi — knotted thin filaments; κi — circulation of γi; u — induced velocity arising from the Biot-Savart integral * This also holds for the cases of thin vortex filaments and thin magnetic flux tubes (Barenghi, Ricca & Samuels, 2001): ω = ω0ˆt, where: ω0 — constant, ˆt — the unit tangent to the vortex axis Motivations Knot polynomials in terms of helicity Examples Outlook Heuristics (. continued) i P H u·dl • Thus, eiS ≡ eiH =⇒ e k γk , and the functional integral turns to be Z i P H u·dl2 [Du] e k γk . i P H u·dl This means the exponential e k γk is at the center of the problem. • Inspiration! This provides us a totally new starting point for studying this problem — the exponential form of the line integral P H i u·dl iH e k γk = e [Remark]: in this classical mechanical case we don’t do functional integrals. • Recall the Edinburgh workshop: • Significance of Line Integral • Phases in the generalized Aharonov-Bohm effect Motivations Knot polynomials in terms of helicity Examples Outlook Knot polynomial invariants in terms of helicity • Main theorem • Revisit to knot polynomials • Realization of knot polynomials in a physical system • Preliminary for proof • Proof of theorem (2 steps) Motivations Knot polynomials in terms of helicity Examples Outlook Result: knot polynomials in terms of helicity For simplicity, let us refer to superfluid vortex lines: • Circulations of vortices are quantized and usually equal, κi = κ, in superfluid. • For thin filaments in quantum fluids, since ω = ω0ˆt, helicity can be reduced to X I H (K) = κi u · dl, u — velocity induced from the Biot-Savart integral. i γi By rescaling eiH(K) → tH(K), and taking κ = 1 for simplicity, we have Theorem (Liu & Ricca, 2012) Let K be a fluid knot or link. If the helicity of K is H = H (K), then H(K) H u·dl t = t K , appropriately re-scaled, satisfies the skein relations of the Jones polynomial V = V (K). Motivations Knot polynomials in terms of helicity Examples Outlook Revisit to knot polynomials • Reidemeister moves: Two isotopic knots can be transformed into each other through a sequence of Reidemeister moves. Figure: Type-I, II & III Reidemeister moves. Ambient-isotopic: invariant under all the three types of moves Regular-isotopic: invariant under only the type-II & III moves • Crossing states at a site: The method of knot theory to determine topological equivalence among knots is to examine the different states at a crossing site. For instance, for oriented links, we consider They represent three links which are almost the same but differ only at the crossing site. Motivations Knot polynomials in terms of helicity Examples Outlook Revisit to knot polynomials (. continued) • A knot polynomial is defined by the skein relations it satisfies. The skein relations can be regarded as the recurrence relations for computing the knot polynomial (see the examples in the next section). • Skein relations of typical knot polynomials • Jones polynomial (for oriented links; ambient-isotopic) V ( ) = 1 −1 1 − 1 τ Vf I − τV @I@ = τ 2 − τ 2 V ] • Kauffman bracket polynomial (for unoriented links; regular-isotopic) h i = 1, f = a h i + a−1 h i , = a−1 h i + a h i , h t i = − a2 + a−2 h i , • HOMFLY polynomialc (for oriented links; ambient-isotopic) P ( ) = 1 t−1Pf I − tP @I@ = zP ] • Alexander-Conway polynomial, R-polynomial, . Motivations Knot polynomials in terms of helicity Examples Outlook Realizing skein relations • In a physical system, to find a knot polynomial means to realize its skein relations. • Example: Guadagnini, et al. (Nucl. Phys. B, 1990) proposed a method to derive skein relations of knot polynomials in a non-abelian Chern-Simons study. Consider the difference between an over- and under-crossing: This is represented by the difference between two VEVs of Wilson loops * !+ * "I # !+ W I − W @I@ = T r ··· A ··· U (3, 4) ··· loop Z 1 = T r ··· i F ··· U (3, 4) ··· area 2 δS δ Then, using a property of Chern-Simons, F = δA , the field tensor F is replaced by the operator δA , and then lead to the desired skein relations. • This is just one way to realize skein relations. For your problems, you can develop a different method to realize the skein relations of a knot polynomial. Motivations Knot polynomials in terms of helicity Examples Outlook Preliminary remark: reduction techniques on crossing sites • Our strategy: P H u·dl To take benefit of the exponential form of the line integral, tH(K) = t i γi , by reforming the strands of K — technique of local path-addition. • Take L+ and L− for instance: We add local imaginary paths to the strands, which have opposite orientations and can cancel each other. This is an identical operation from the fluid mechanical point of view.
Recommended publications
  • The Borromean Rings: a Video About the New IMU Logo

    The Borromean Rings: a Video About the New IMU Logo

    The Borromean Rings: A Video about the New IMU Logo Charles Gunn and John M. Sullivan∗ Technische Universitat¨ Berlin Institut fur¨ Mathematik, MA 3–2 Str. des 17. Juni 136 10623 Berlin, Germany Email: {gunn,sullivan}@math.tu-berlin.de Abstract This paper describes our video The Borromean Rings: A new logo for the IMU, which was premiered at the opening ceremony of the last International Congress. The video explains some of the mathematics behind the logo of the In- ternational Mathematical Union, which is based on the tight configuration of the Borromean rings. This configuration has pyritohedral symmetry, so the video includes an exploration of this interesting symmetry group. Figure 1: The IMU logo depicts the tight config- Figure 2: A typical diagram for the Borromean uration of the Borromean rings. Its symmetry is rings uses three round circles, with alternating pyritohedral, as defined in Section 3. crossings. In the upper corners are diagrams for two other three-component Brunnian links. 1 The IMU Logo and the Borromean Rings In 2004, the International Mathematical Union (IMU), which had never had a logo, announced a competition to design one. The winning entry, shown in Figure 1, was designed by one of us (Sullivan) with help from Nancy Wrinkle. It depicts the Borromean rings, not in the usual diagram (Figure 2) but instead in their tight configuration, the shape they have when tied tight in thick rope. This IMU logo was unveiled at the opening ceremony of the International Congress of Mathematicians (ICM 2006) in Madrid. We were invited to produce a short video [10] about some of the mathematics behind the logo; it was shown at the opening and closing ceremonies, and can be viewed at www.isama.org/jms/Videos/imu/.
  • Knots: a Handout for Mathcircles

    Knots: a Handout for Mathcircles

    Knots: a handout for mathcircles Mladen Bestvina February 2003 1 Knots Informally, a knot is a knotted loop of string. You can create one easily enough in one of the following ways: • Take an extension cord, tie a knot in it, and then plug one end into the other. • Let your cat play with a ball of yarn for a while. Then find the two ends (good luck!) and tie them together. This is usually a very complicated knot. • Draw a diagram such as those pictured below. Such a diagram is a called a knot diagram or a knot projection. Trefoil and the figure 8 knot 1 The above two knots are the world's simplest knots. At the end of the handout you can see many more pictures of knots (from Robert Scharein's web site). The same picture contains many links as well. A link consists of several loops of string. Some links are so famous that they have names. For 2 2 3 example, 21 is the Hopf link, 51 is the Whitehead link, and 62 are the Bor- romean rings. They have the feature that individual strings (or components in mathematical parlance) are untangled (or unknotted) but you can't pull the strings apart without cutting. A bit of terminology: A crossing is a place where the knot crosses itself. The first number in knot's \name" is the number of crossings. Can you figure out the meaning of the other number(s)? 2 Reidemeister moves There are many knot diagrams representing the same knot. For example, both diagrams below represent the unknot.
  • Dehn Filling of the “Magic” 3-Manifold

    Dehn Filling of the “Magic” 3-Manifold

    communications in analysis and geometry Volume 14, Number 5, 969–1026, 2006 Dehn filling of the “magic” 3-manifold Bruno Martelli and Carlo Petronio We classify all the non-hyperbolic Dehn fillings of the complement of the chain link with three components, conjectured to be the smallest hyperbolic 3-manifold with three cusps. We deduce the classification of all non-hyperbolic Dehn fillings of infinitely many one-cusped and two-cusped hyperbolic manifolds, including most of those with smallest known volume. Among other consequences of this classification, we mention the following: • for every integer n, we can prove that there are infinitely many hyperbolic knots in S3 having exceptional surgeries {n, n +1, n +2,n+3}, with n +1,n+ 2 giving small Seifert manifolds and n, n + 3 giving toroidal manifolds. • we exhibit a two-cusped hyperbolic manifold that contains a pair of inequivalent knots having homeomorphic complements. • we exhibit a chiral 3-manifold containing a pair of inequivalent hyperbolic knots with orientation-preservingly homeomorphic complements. • we give explicit lower bounds for the maximal distance between small Seifert fillings and any other kind of exceptional filling. 0. Introduction We study in this paper the Dehn fillings of the complement N of the chain link with three components in S3, shown in figure 1. The hyperbolic structure of N was first constructed by Thurston in his notes [28], and it was also noted there that the volume of N is particularly small. The relevance of N to three-dimensional topology comes from the fact that by filling N, one gets most of the hyperbolic manifolds known and most of the interesting non-hyperbolic fillings of cusped hyperbolic manifolds.
  • A Remarkable 20-Crossing Tangle Shalom Eliahou, Jean Fromentin

    A Remarkable 20-Crossing Tangle Shalom Eliahou, Jean Fromentin

    A remarkable 20-crossing tangle Shalom Eliahou, Jean Fromentin To cite this version: Shalom Eliahou, Jean Fromentin. A remarkable 20-crossing tangle. 2016. hal-01382778v2 HAL Id: hal-01382778 https://hal.archives-ouvertes.fr/hal-01382778v2 Preprint submitted on 16 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A REMARKABLE 20-CROSSING TANGLE SHALOM ELIAHOU AND JEAN FROMENTIN Abstract. For any positive integer r, we exhibit a nontrivial knot Kr with r− r (20·2 1 +1) crossings whose Jones polynomial V (Kr) is equal to 1 modulo 2 . Our construction rests on a certain 20-crossing tangle T20 which is undetectable by the Kauffman bracket polynomial pair mod 2. 1. Introduction In [6], M. B. Thistlethwaite gave two 2–component links and one 3–component link which are nontrivial and yet have the same Jones polynomial as the corre- sponding unlink U 2 and U 3, respectively. These were the first known examples of nontrivial links undetectable by the Jones polynomial. Shortly thereafter, it was shown in [2] that, for any integer k ≥ 2, there exist infinitely many nontrivial k–component links whose Jones polynomial is equal to that of the k–component unlink U k.
  • Forming the Borromean Rings out of Polygonal Unknots

    Forming the Borromean Rings out of Polygonal Unknots

    Forming the Borromean Rings out of polygonal unknots Hugh Howards 1 Introduction The Borromean Rings in Figure 1 appear to be made out of circles, but a result of Freedman and Skora shows that this is an optical illusion (see [F] or [H]). The Borromean Rings are a special type of Brunninan Link: a link of n components is one which is not an unlink, but for which every sublink of n − 1 components is an unlink. There are an infinite number of distinct Brunnian links of n components for n ≥ 3, but the Borromean Rings are the most famous example. This fact that the Borromean Rings cannot be formed from circles often comes as a surprise, but then we come to the contrasting result that although it cannot be built out of circles, the Borromean Rings can be built out of convex curves, for example, one can form it from two circles and an ellipse. Although it is only one out of an infinite number of Brunnian links of three Figure 1: The Borromean Rings. 1 components, it is the only one which can be built out of convex compo- nents [H]. The convexity result is, in fact a bit stronger and shows that no 4 component Brunnian link can be made out of convex components and Davis generalizes this result to 5 components in [D]. While this shows it is in some sense hard to form most Brunnian links out of certain shapes, the Borromean Rings leave some flexibility. This leads to the question of what shapes can be used to form the Borromean Rings and the following surprising conjecture of Matthew Cook at the California Institute of Technology.
  • Hyperbolic Structures from Link Diagrams

    Hyperbolic Structures from Link Diagrams

    University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2012 Hyperbolic Structures from Link Diagrams Anastasiia Tsvietkova [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Geometry and Topology Commons Recommended Citation Tsvietkova, Anastasiia, "Hyperbolic Structures from Link Diagrams. " PhD diss., University of Tennessee, 2012. https://trace.tennessee.edu/utk_graddiss/1361 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Anastasiia Tsvietkova entitled "Hyperbolic Structures from Link Diagrams." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Mathematics. Morwen B. Thistlethwaite, Major Professor We have read this dissertation and recommend its acceptance: Conrad P. Plaut, James Conant, Michael Berry Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) Hyperbolic Structures from Link Diagrams A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville Anastasiia Tsvietkova May 2012 Copyright ©2012 by Anastasiia Tsvietkova. All rights reserved. ii Acknowledgements I am deeply thankful to Morwen Thistlethwaite, whose thoughtful guidance and generous advice made this research possible.
  • Introduction to Vassiliev Knot Invariants First Draft. Comments

    Introduction to Vassiliev Knot Invariants First Draft. Comments

    Introduction to Vassiliev Knot Invariants First draft. Comments welcome. July 20, 2010 S. Chmutov S. Duzhin J. Mostovoy The Ohio State University, Mansfield Campus, 1680 Univer- sity Drive, Mansfield, OH 44906, USA E-mail address: [email protected] Steklov Institute of Mathematics, St. Petersburg Division, Fontanka 27, St. Petersburg, 191011, Russia E-mail address: [email protected] Departamento de Matematicas,´ CINVESTAV, Apartado Postal 14-740, C.P. 07000 Mexico,´ D.F. Mexico E-mail address: [email protected] Contents Preface 8 Part 1. Fundamentals Chapter 1. Knots and their relatives 15 1.1. Definitions and examples 15 § 1.2. Isotopy 16 § 1.3. Plane knot diagrams 19 § 1.4. Inverses and mirror images 21 § 1.5. Knot tables 23 § 1.6. Algebra of knots 25 § 1.7. Tangles, string links and braids 25 § 1.8. Variations 30 § Exercises 34 Chapter 2. Knot invariants 39 2.1. Definition and first examples 39 § 2.2. Linking number 40 § 2.3. Conway polynomial 43 § 2.4. Jones polynomial 45 § 2.5. Algebra of knot invariants 47 § 2.6. Quantum invariants 47 § 2.7. Two-variable link polynomials 55 § Exercises 62 3 4 Contents Chapter 3. Finite type invariants 69 3.1. Definition of Vassiliev invariants 69 § 3.2. Algebra of Vassiliev invariants 72 § 3.3. Vassiliev invariants of degrees 0, 1 and 2 76 § 3.4. Chord diagrams 78 § 3.5. Invariants of framed knots 80 § 3.6. Classical knot polynomials as Vassiliev invariants 82 § 3.7. Actuality tables 88 § 3.8. Vassiliev invariants of tangles 91 § Exercises 93 Chapter 4.
  • Rock-Paper-Scissors Meets Borromean Rings

    Rock-Paper-Scissors Meets Borromean Rings

    ROCK-PAPER-SCISSORS MEETS BORROMEAN RINGS MARC CHAMBERLAND AND EUGENE A. HERMAN 1. Introduction Directed graphs with an odd number of vertices n, where each vertex has both (n − 1)/2 incoming and outgoing edges, have a rich structure. We were lead to their study by both the Borromean rings and the game rock- paper-scissors. An interesting interplay between groups, graphs, topological links, and matrices reveals the structure of these objects, and for larger values of n, extensive computation produces some surprises. Perhaps most surprising is how few of the larger graphs have any symmetry and those with symmetry possess very little. In the final section, we dramatically sped up the computation by first computing a “profile” for each graph. 2. Three Weapons Let’s start with the two-player game rock-paper-scissors or RPS(3). The players simultaneously put their hands in one of three positions: rock (fist), paper (flat palm), or scissors (fist with the index and middle fingers sticking out). The winner of the game is decided as follows: paper covers rock, rock smashes scissors, and scissors cut paper. Mathematically, this game is referred to as a balanced tournament: with an odd number n of weapons, each weapon beats (n−1)/2 weapons and loses to the same number. This mutual dominance/submission connects RPS(3) with a seemingly disparate object: Borromean rings. Figure 1. Borromean Rings. 1 2 MARCCHAMBERLANDANDEUGENEA.HERMAN The Borromean rings in Figure 1 consist of three unknots in which the red ring lies on top of the blue ring, the blue on top of the green, and the green on top of the red.
  • Alternating Knots

    Alternating Knots

    ALTERNATING KNOTS WILLIAM W. MENASCO Abstract. This is a short expository article on alternating knots and is to appear in the Concise Encyclopedia of Knot Theory. Introduction Figure 1. P.G. Tait's first knot table where he lists all knot types up to 7 crossings. (From reference [6], courtesy of J. Hoste, M. Thistlethwaite and J. Weeks.) 3 ∼ A knot K ⊂ S is alternating if it has a regular planar diagram DK ⊂ P(= S2) ⊂ S3 such that, when traveling around K , the crossings alternate, over-under- over-under, all the way along K in DK . Figure1 show the first 15 knot types in P. G. Tait's earliest table and each diagram exhibits this alternating pattern. This simple arXiv:1901.00582v1 [math.GT] 3 Jan 2019 definition is very unsatisfying. A knot is alternating if we can draw it as an alternating diagram? There is no mention of any geometric structure. Dissatisfied with this characterization of an alternating knot, Ralph Fox (1913-1973) asked: "What is an alternating knot?" black white white black Figure 2. Going from a black to white region near a crossing. 1 2 WILLIAM W. MENASCO Let's make an initial attempt to address this dissatisfaction by giving a different characterization of an alternating diagram that is immediate from the over-under- over-under characterization. As with all regular planar diagrams of knots in S3, the regions of an alternating diagram can be colored in a checkerboard fashion. Thus, at each crossing (see figure2) we will have \two" white regions and \two" black regions coming together with similarly colored regions being kitty-corner to each other.
  • The Jones Polynomial Through Linear Algebra

    The Jones Polynomial Through Linear Algebra

    The Jones polynomial through linear algebra Iain Moffatt University of South Alabama Workshop in Knot Theory Waterloo, 24th September 2011 I. Moffatt (South Alabama) UW 2011 1 / 39 What and why What we’ll see The construction of link invariants through R-matrices. (c.f. Reshetikhin-Turaev invariants, quantum invariants) Why this? Can do some serious math using material from Linear Algebra 1. Illustrates how math works in the wild: start with a problem you want to solve; figure out an easier problem that you can solve; build up from this to solve your original problem. See the interplay between algebra, combinatorics and topology! It’s my favourite bit of math! I. Moffatt (South Alabama) UW 2011 2 / 39 What we’re trying to do A knot is a circle, S1, sitting in 3-space R3. A link is a number of disjoint circles in 3-space R3. Knots and links are considered up to isotopy. This means you can “move then round in space, but you can’t cut or glue them”. I. Moffatt (South Alabama) UW 2011 3 / 39 What we’re trying to do A knot is a circle, S1, sitting in 3-space R3. A link is a number of disjoint circles in 3-space R3. Knots and links are considered up to isotopy. This means you can “move then round in space, but you can’t cut or glue them”. The fundamental problem in knot theory is to determine whether or not two links are isotopic. =? =? I. Moffatt (South Alabama) UW 2011 3 / 39 To do this we need knot invariants: F : Links (a set) such (Isotopy) ! that F(L) = F(L0) = L = L0, Aim: construct6 link invariants) 6 using linear algebra.
  • Knot Theory and the Alexander Polynomial

    Knot Theory and the Alexander Polynomial

    Knot Theory and the Alexander Polynomial Reagin Taylor McNeill Submitted to the Department of Mathematics of Smith College in partial fulfillment of the requirements for the degree of Bachelor of Arts with Honors Elizabeth Denne, Faculty Advisor April 15, 2008 i Acknowledgments First and foremost I would like to thank Elizabeth Denne for her guidance through this project. Her endless help and high expectations brought this project to where it stands. I would Like to thank David Cohen for his support thoughout this project and through- out my mathematical career. His humor, skepticism and advice is surely worth the $.25 fee. I would also like to thank my professors, peers, housemates, and friends, particularly Kelsey Hattam and Katy Gerecht, for supporting me throughout the year, and especially for tolerating my temporary insanity during the final weeks of writing. Contents 1 Introduction 1 2 Defining Knots and Links 3 2.1 KnotDiagramsandKnotEquivalence . ... 3 2.2 Links, Orientation, and Connected Sum . ..... 8 3 Seifert Surfaces and Knot Genus 12 3.1 SeifertSurfaces ................................. 12 3.2 Surgery ...................................... 14 3.3 Knot Genus and Factorization . 16 3.4 Linkingnumber.................................. 17 3.5 Homology ..................................... 19 3.6 TheSeifertMatrix ................................ 21 3.7 TheAlexanderPolynomial. 27 4 Resolving Trees 31 4.1 Resolving Trees and the Conway Polynomial . ..... 31 4.2 TheAlexanderPolynomial. 34 5 Algebraic and Topological Tools 36 5.1 FreeGroupsandQuotients . 36 5.2 TheFundamentalGroup. .. .. .. .. .. .. .. .. 40 ii iii 6 Knot Groups 49 6.1 TwoPresentations ................................ 49 6.2 The Fundamental Group of the Knot Complement . 54 7 The Fox Calculus and Alexander Ideals 59 7.1 TheFreeCalculus ...............................
  • How Can We Say 2 Knots Are Not the Same?

    How Can We Say 2 Knots Are Not the Same?

    How can we say 2 knots are not the same? SHRUTHI SRIDHAR What’s a knot? A knot is a smooth embedding of the circle S1 in IR3. A link is a smooth embedding of the disjoint union of more than one circle Intuitively, it’s a string knotted up with ends joined up. We represent it on a plane using curves and ‘crossings’. The unknot A ‘figure-8’ knot A ‘wild’ knot (not a knot for us) Hopf Link Two knots or links are the same if they have an ambient isotopy between them. Representing a knot Knots are represented on the plane with strands and crossings where 2 strands cross. We call this picture a knot diagram. Knots can have more than one representation. Reidemeister moves Operations on knot diagrams that don’t change the knot or link Reidemeister moves Theorem: (Reidemeister 1926) Two knot diagrams are of the same knot if and only if one can be obtained from the other through a series of Reidemeister moves. Crossing Number The minimum number of crossings required to represent a knot or link is called its crossing number. Knots arranged by crossing number: Knot Invariants A knot/link invariant is a property of a knot/link that is independent of representation. Trivial Examples: • Crossing number • Knot Representations / ~ where 2 representations are equivalent via Reidemester moves Tricolorability We say a knot is tricolorable if the strands in any projection can be colored with 3 colors such that every crossing has 1 or 3 colors and or the coloring uses more than one color.