DOCSLIB.ORG

Braid and Knot Theory in Dimension Four, 2002 94 Mara D

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Braid and Knot Theory in Dimension Four, 2002 94 Mara D http://dx.doi.org/10.1090/surv/095 Titles in This Series 95 Seiichi Kamada, Braid and knot theory in dimension four, 2002 94 Mara D. Neusel and Larry Smith, Invariant theory of finite groups, 2002 93 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 2: Model operators and systems, 2002 92 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 1: Hardy, Hankel, and Toeplitz, 2002 91 Richard Montgomery, A tour of subriemannian geometries, their geodesies and applications, 2002 90 Christian Gerard and Izabella Laba, Multiparticle quantum scattering in constant magnetic fields, 2002 89 Michel Ledoux, The concentration of measure phenomenon, 2001 88 Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, 2001 87 Bruno Poizat, Stable groups, 2001 86 Stanley N. Burris, Number theoretic density and logical limit laws, 2001 85 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, 2001 84 Laszlo Fuchs and Luigi Salce, Modules over non-Noetherian domains, 2001 83 Sigurdur Helgason, Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, 2000 82 Goro Shimura, Arithmeticity in the theory of automorphic forms, 2000 81 Michael E. Taylor, Tools for PDE: Pseudodifferential operators, paradifferential operators, and layer potentials, 2000 80 Lindsay N. Childs, Taming wild extensions: Hopf algebras and local Galois module theory, 2000 79 Joseph A. Cima and William T. Ross, Trie backward shift on the Hardy space, 2000 78 Boris A. Kupershmidt, KP or mKP: Noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems, 2000 77 Fumio Hiai and Denes Petz, The semicircle law, free random variables and entropy, 2000 76 Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmuller theory, 2000 75 Greg Hjorth, Classification and orbit equivalence relations, 2000 74 Daniel W. Stroock, An introduction to the analysis of paths on a Riemannian manifold, 2000 73 John Locker, Spectral theory of non-self-adjoint two-point differential operators, 2000 72 Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, 1999 71 Lajos Pukanszky, Characters of connected Lie groups, 1999 70 Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems and differential equations, 1999 69 C. T. C. Wall (A. A. Ranicki, Editor), Surgery on compact manifolds, second edition, 1999 68 David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, 1999 67 A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, second edition, 2000 66 Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, 1999 65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra, 1999 For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/. Mathematical Surveys and Monographs Volume 95 Braid and Knot Theory in Dimension Four Seiichi Kamada American Mathematical Society WDED* Editorial Board Peter Landweber Tudor Ratiu Michael Loss, Chair J. T. Stafford 2000 Mathematics Subject Classification. Primary 57Q45; Secondary 20F36, 57M05, 57M12, 57M25, 57Q35. ABSTRACT. Braid theory and knot theory are related to each other via two famous results due to Alexander and Markov. Alexander's theorem states that any knot or link can be put into braid form. Markov's theorem gives necessary and sufficient conditions to conclude that two braids represent the same knot or link. Thus one can use braid theory to study knot theory, and vice versa. In this book we generalize braid theory to dimension four. We develop the theory of surface braids and apply it to study surface links. Especially, the generalized Alexander and Markov theorems in dimension four are given. This book is the first place that contains a complete proof of the generalized Markov theorem. Surface links are also studied via the motion picture method, and some important techniques of this method are studied. For surface braids, various methods to describe them are introduced and developed: the motion picture method, the chart description, the braid monodromy, and the braid system. These tools are fundamental to understanding and computing invariants of surface braids and surface links. A table of knotted surfaces is included with a computation of Alexander polynomials. The braid techniques are extended to represent link homotopy classes. Library of Congress Cataloging-in-Publication Data Kamada, Seiichi, 1964- Braid and knot theory in dimension four / Seiichi Kamada. p. cm. — (Mathematical surveys and monographs ; v. 95) Includes bibliographical references and index. ISBN 0-8218-2969-6 (alk. paper) 1. Braid theory. 2. Knot theory. I. Title. II. Mathematical surveys and monographs ; no. 95. QA612.23.K36 2002 514/.224—dc21 2002018274 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionOams.org. © 2002 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 02 Contents Preface xi Chapter 0. Basic Notions and Notation 1 0.1. Properness and Local Flatness 1 0.2. Isotopy 2 0.3. Ambient Isotopy 3 0.4. Notation 3 Part 1. Classical Braids and Links 5 Chapter 1. Braids 7 1.1. Geometric Braids and the Braid Group 7 1.2. Configuration Space 8 1.3. Pure Braids and Braid Permutation 10 1.4. Brick Regular Neighborhood 11 1.5. Braid Isotopy Extension Theorem 12 1.6. Polygonal Braids 13 1.7. Braid Diagrams 14 1.8. Presentation of the Braid Group 15 Chapter 2. Braid Automorphisms 19 2.1. The Isotopies Associated with a Braid 19 2.2. Mapping Class Group 20 2.3. Hurwitz Arc System 21 2.4. Hurwitz Arc System of a Braid 22 2.5. Injectivity of $ 23 2.6. Artin's Braid Automorphism 23 2.7. Slide Action of the Braid Group 25 Chapter 3. Classical Links 27 3.1. Knots and Links 27 3.2. Basic Symmetries 28 3.3. The Regular Neighborhood of a Link 28 3.4. Trivial Links 29 3.5. Split Union and Connected Sum 30 3.6. Combinatorial Equivalence 30 3.7. Regular Projections 31 3.8. Link Diagrams 32 3.9. Reidemeister Moves 33 3.10. The Group of a Link 35 vi CONTENTS 3.11. A Note on Knots as Embeddings 38 Chapter 4. Braid Presentation of Links 41 4.1. Presenting Links by Braids 41 4.2. The Braiding Process 42 4.3. Markov's Theorem 43 Chapter 5. Deformation Chain and Markov's Theorem 47 5.1. £ Operation 47 5.2. Deformation Chains 48 5.3. Deformation Chains at the Triangulation Level 49 5.4. Height Reduction and Markov's Theorem 50 5.5. Notes 50 Part 2. Surface Knots and Links 51 Chapter 6. Surface Links 53 6.1. Surface Links 53 6.2. Trivial Surface Links 53 6.3. Combinatorial Equivalence 55 Chapter 7. Surface Link Diagrams 57 7.1. Generic Maps 57 7.2. Surface Link Diagrams 58 7.3. Elementary Moves to Diagrams 59 Chapter 8. Motion Pictures 63 8.1. Motion Pictures 63 8.2. Motion Pictures of Surface Links 63 8.3. The Motion Picture of a Non-locally Flat Surface 65 8.4. Elementary Critical Bands 66 8.5. Trivial Disk Systems 67 8.6. Proof of Lemma 8.7 69 Chapter 9. Normal Forms of Surface Links 71 9.1. Link Transformation Sequence 71 9.2. The Realizing Surface 72 9.3. Technical Lemmas on Realizing Surfaces 72 9.4. Closed Realizing Surface 75 9.5. The Normal Form 75 Chapter 10. Examples (Spinning) 79 10.1. Spinning Construction 79 10.2. Twist-Spinning Construction 80 10.3. Deform-Spinning Construction 82 10.4. Deformations 83 10.5. Deform-Spun Projective Planes 84 Chapter 11. Ribbon Surface Links 87 11.1. 1-Handle and 2-Handle Surgery 87 11.2. Ribbon Surface Links 88 CONTENTS vii 11.3. Slice versus Ribbon 89 Chapter 12. Presentations of Surface Link Groups 91 12.1. The Presentation from a Diagram 91 12.2. The Presentation from a Motion Picture 92 12.3. The Elementary Ideals 94 Part 3. Surface Braids 97 Chapter 13. Branched Coverings 99 13.1. Branched Coverings 99 13.2. Types of Branch Points 100 13.3. Riemann-Hurwitz Formula 101 13.4. Monodromy 102 13.5. Simple Branched Coverings 103 Chapter 14. Surface Braids 105 14.1. Surface Braids 105 14.2. Motion Pictures 105 14.3. Trivial Surface Braids 107 14.4. Simple Surface Braids 108 14.5. Equivalence Relations 109 Chapter 15. Products of Surface Braids 113 15.1. Products of Motion Pictures 113 15.2. Products of Surface Braids 114 15.3. The Surface Braid Monoid 114 Chapter 16. Braided Surfaces 117 16.1. Braided Surfaces 117 16.2. The Motion Picture of a Braided Surface 117 16.3. Equivalence Relations on Braided Surfaces 117 16.4.
Load more

Recommended publications
  • Slicing Bing Doubles
    Algebraic & Geometric Topology 6 (2006) 3001–999 3001 Slicing Bing doubles DAVID CIMASONI Bing doubling is an operation which produces a 2–component boundary link B.K/ from a knot K. If K is slice, then B.K/ is easily seen to be boundary slice. In this paper, we investigate whether the converse holds. Our main result is that if B.K/ is boundary slice, then K is algebraically slice. We also show that the Rasmussen invariant can tell that certain Bing doubles are not smoothly slice. 57M25; 57M27 Introduction Bing doubling [2] is a standard construction which, given a knot K in S 3 , produces a 2–component oriented link B.K/ as illustrated below. (See Section 1 for a precise definition.) K B.K/ Figure 1: The figure eight knot and its Bing double It is easy to check that if the knot K is slice, then the link B.K/ is slice. Does the converse hold ? An affirmative answer to this question seems out of reach. However, a result obtained independently by S Harvey [9] and P Teichner [22] provides a first step in this direction. 1 Recall that the Levine–Tristram signature of a knot K is the function K S ޚ 1 W ! defined as follows: for ! S , K .!/ is given by the signature of the Hermitian 2 matrix .1 !/A .1 !/A , where A is a Seifert matrix for K and A denotes the C transposed matrix. Published: December 2006 DOI: 10.2140/agt.2006.6.3001 3002 David Cimasoni Theorem (Harvey [9], Teichner [22]) If B.K/ is slice, then the integral over S 1 of the Levine–Tristram signature of K is zero.
    [Show full text]
  • (1,1)-Decompositions of Rational Pretzel Knots
    East Asian Math. J. Vol. 32 (2016), No. 1, pp. 077{084 http://dx.doi.org/10.7858/eamj.2016.008 (1,1)-DECOMPOSITIONS OF RATIONAL PRETZEL KNOTS Hyun-Jong Song* Abstract. We explicitly derive diagrams representing (1,1)-decompositions of rational pretzel knots Kβ = M((−2; 1); (3; 1); (j6β + 1j; β)) from four unknotting tunnels for β = 1; −2 and 2. 1. Preliminaries Let V be an unknotted solid torus and let α be a properly embedded arc in V . We say that α is trivial in V if and only if there exists an arc β in @V joining the two end points of α , and a 2-disk D in V such that the boundary @D of D is a union of α and β. Such a disk D is called a cancelling disk of α. Equivalently α is trivial in V if and only if H = V − N(α)◦, the complement of the open tubular neighbourhood N(α) of α in V is a handlebody of genus 2. Note that a cancelling disk of the trivial arc α and a meridian disk of V disjoint from α contribute a meridian disk system of the handlebody H A knot K in S3 is referred to as admitting genus 1, 1-bridge decomposition 3 or (1,1)-decomposition for short if and only if (S ;K) is split into pairs (Vi; αi) (i = 1; 2) of trivial arcs in solid tori determined by a Heegaard torus T = @Vi of S3. Namely we have: 3 (S ;K) = (V1; α1) [T (V2; α2) .
    [Show full text]
  • Arxiv:1208.5857V2 [Math.GT] 27 Nov 2012 Hscs Ecncnie Utetspace Quotient a Consider Can We Case This Hoe 1.1
    A GOOD PRESENTATION OF (−2, 3, 2s + 1)-TYPE PRETZEL KNOT GROUP AND R-COVERED FOLIATION YASUHARU NAKAE Abstract. Let Ks bea(−2, 3, 2s+1)-type Pretzel knot (s ≧ 3) and EKs (p/q) be a closed manifold obtained by Dehn surgery along Ks with a slope p/q. We prove that if q > 0, p/q ≧ 4s + 7 and p is odd, then EKs (p/q) cannot contain an R-covered foliation. This result is an extended theorem of a part of works of Jinha Jun for (−2, 3, 7)-Pretzel knot. 1. Introduction In this paper, we will discuss non-existence of R-covered foliations on a closed 3-manifold obtained by Dehn surgery along some class of a Pretzel knot. A codimension one, transversely oriented foliation F on a closed 3-manifold M is called a Reebless foliation if F does not contain a Reeb component. By the theorems of Novikov [13], Rosenberg [17], and Palmeira [14], if M is not homeo- morphic to S2 × S1 and contains a Reebless foliation, then M has properties that the fundamental group of M is infinite, the universal cover M is homeomorphic to R3 and all leaves of its lifted foliation F on M are homeomorphic to a plane. In this case we can consider a quotient space T = M/F, and Tfis called a leaf space of F. The leaf space T becomes a simplye connectedf 1-manifold, but it might be a non-Hausdorff space. If the leaf space is homeomorphicf e to R, F is called an R- covered foliation.
    [Show full text]
  • Hollow Braid Eye Splice
    The Back Splice A properly sized hollow braid splicing fid will make this splice easier. Hollow braid splices must have the opposing core tucked in at least eight inches when finished. Use discretionary thinking when determining whether or not to apply a whipping to the back splice on hollow braid ropes. 5/16” ¼” 3/16” 3/8” Whipping Twine Hollow Braid Appropriate Sized Knife Splicing Fids STEP ONE: The first step with FIG. 1 most hollow braid splices involve inserting the end of the rope into the hollow end of an appropriately sized splicing fid (Figures 1 & 2). Fids are sized according to the diameter of the rope. A 3/8” diameter rope will be used in this demonstration, therefore a 3/8” fid is the appropriate size. FIG. 2 The fid can prove useful when estimating the length the opposing core is tucked. A minimum tuck of eight inches is required. FIG. 2A STEP TWO: After inserting the end of the rope into a splicing fid (figure 2A) – Loosen the braid in the rope FIG. 3 approximately 10” to 12” from the end to be spliced (figure 3). Approximately 10” to 12” From the end of the rope. Push the pointed end of the fid into one of the openings of the braid, allowing the fid to travel down the hollow center of the braided rope (figures 4 & 5). FIG. 4 FIG. 5 FIG.6 STEP THREE: Allow the fid to travel down the hollow center of the braided rope 8” or more. Compressing the rope on the fid will allow a distance safely in excess of 8” (figure 6).
    [Show full text]
  • Ribbon Concordances and Doubly Slice Knots
    Ribbon concordances and doubly slice knots Ruppik, Benjamin Matthias Advisors: Arunima Ray & Peter Teichner The knot concordance group Glossary Definition 1. A knot K ⊂ S3 is called (topologically/smoothly) slice if it arises as the equatorial slice of a (locally flat/smooth) embedding of a 2-sphere in S4. – Abelian invariants: Algebraic invari- Proposition. Under the equivalence relation “K ∼ J iff K#(−J) is slice”, the commutative monoid ants extracted from the Alexander module Z 3 −1 A (K) = H1( \ K; [t, t ]) (i.e. from the n isotopy classes of o connected sum S Z K := 1 3 , # := oriented knots S ,→S of knots homology of the universal abelian cover of the becomes a group C := K/ ∼, the knot concordance group. knot complement). The neutral element is given by the (equivalence class) of the unknot, and the inverse of [K] is [rK], i.e. the – Blanchfield linking form: Sesquilin- mirror image of K with opposite orientation. ear form on the Alexander module ( ) sm top Bl: AZ(K) × AZ(K) → Q Z , the definition Remark. We should be careful and distinguish the smooth C and topologically locally flat C version, Z[Z] because there is a huge difference between those: Ker(Csm → Ctop) is infinitely generated! uses that AZ(K) is a Z[Z]-torsion module. – Levine-Tristram-signature: For ω ∈ S1, Some classical structure results: take the signature of the Hermitian matrix sm top ∼ ∞ ∞ ∞ T – There are surjective homomorphisms C → C → AC, where AC = Z ⊕ Z2 ⊕ Z4 is Levine’s algebraic (1 − ω)V + (1 − ω)V , where V is a Seifert concordance group.
    [Show full text]
  • Splicing Guide
    SPLICING GUIDE EN SPLICING GUIDE SPLICING GUIDE Contents Splicing Guide General Splicing 3 General Splicing Tips Tools Required Fid Lengths 3 1. Before starting, it is a good idea to read through the – Masking Tape – Sharp Knife directions so you understand the general concepts and – Felt Tip Marker – Measuring Tape Single Braid 4 principles of the splice. – Splicing Fide 2. A “Fid” length equals 21 times the diameter of the rope Single Braid Splice (Bury) 4 (Ref Fid Chart). Single Braid Splice (Lock Stitch) 5 3. A “Pic” is the V-shaped strand pairs you see as you look Single Braid Splice (Tuck) 6 down the rope. Double Braid 8 Whipping Rope Handling Double Braid Splice 8 Core-To-Core Splice 11 Seize by whipping or stitching the splice to prevent the cross- Broom Sta-Set X/PCR Splice 13 over from pulling out under the unbalanced load. To cross- Handle stitch, mark off six to eight rope diameters from throat in one rope diameter increments (stitch length). Using same material Tapering the Cover on High-Tech Ropes 15 as cover braid if available, or waxed whipping thread, start at bottom leaving at least eight inches of tail exposed for knotting and work toward the eye where you then cross-stitch work- To avoid kinking, coil rope Pull rope from ing back toward starting point. Cut off thread leaving an eight in figure eight for storage or reel directly, Tapered 8 Plait to Chain Splice 16 inch length and double knot as close to rope as possible. Trim take on deck.
    [Show full text]
  • Complete Rope Splicing Guide (PDF)
    NEW ENGLAND ROPES SPLICING GUIDE NEW ENGLAND ROPES SPLICING GUIDE TABLE OF CONTENTS General - Splicing Fid Lengths 3 Single Braid Eye Splice (Bury) 4 Single Braid Eye Splice (Lock Stitch) 5 Single Braid Eye Splice (Tuck) 6 Double Braid Eye Splice 8 Core-to-Core Eye Splice 11 Sta-Set X/PCR Eye Splice 13 Tachyon Splice 15 Braided Safety Blue & Hivee Eye Splice 19 Tapering the Cover on High-Tech Ropes 21 Mega Plait to Chain Eye Splice 22 Three Strand Rope to Chain Splice 24 Eye Splice (Standard and Tapered) 26 FULL FID LENGTH SHORT FID SECTION LONG FID SECTION 1/4” 5/16” 3/8” 7/16” 1/2” 9/16” 5/8” 2 NEW ENGLAND ROPES SPLICING GUIDE GENERAL-SPLICING TIPS TOOLS REQUIRED 1. Before starting, it is a good idea to read through the directions so you . Masking Tape . Sharp Knife understand the general concepts and principles of the splice. Felt Tip Marker . Measuring Tape 2. A “Fid” length equals 21 times the diameter of the rope (Ref Fid Chart). Splicing Fids 3. A “Pic” is the V-shaped strand pairs you see as you look down the rope. WHIPPING ROPE HANDLING Seize by whipping or stitching the splice to prevent the crossover from Broom pulling out under the unbalanced load. To cross-stitch, mark off six to Handle eight rope diameters from throat in one rope diameter increments (stitch length). Using same material as cover braid if available, or waxed whip- ping thread, start at bottom leaving at least eight inches of tail exposed for knotting and work toward the eye where you then cross-stitch working Pull rope from back toward starting point.
    [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]
  • Directions for Knots: Reef, Bowline, and the Figure Eight
    Directions for Knots: Reef, Bowline, and the Figure Eight Materials Two ropes, each with a blue end and a red end (try masking tape around the ends and coloring them with markers, or using red and blue electrical tape around the ends.) Reef Knot (square knot) 1. Hold the red end of the rope in your left hand and the blue end in your right. 2. Cross the red end over the blue end to create a loop. 3. Pass the red end under the blue end and up through the loop. 4. Pull, but not too tight (leave a small loop at the base of your knot). 5. Hold the red end in your right hand and the blue end in your left. 6. Cross the red end over and under the blue end and up through the loop (here, you are repeating steps 2 and 3) 7. Pull Tight! Bowline The bowline knot (pronounced “bow-lin”) is a loop knot, which means that it is tied around an object or tied when a temporary loop is needed. On USS Constitution, sailors used bowlines to haul heavy loads onto the ship. 1. Hold the blue end of the rope in your left hand and the red end in your right. 2. Cross the red end over the blue end to make a loop. 3. Tuck the red end up and through the loop (pull, but not too tight!). 4. Keep the blue end of the rope in your left hand and the red in your right. 5. Pass the red end behind and around the blue end.
    [Show full text]
  • Fibered Ribbon Disks
    FIBERED RIBBON DISKS KYLE LARSON AND JEFFREY MEIER Abstract. We study the relationship between fibered ribbon 1{knots and fibered ribbon 2{knots by studying fibered slice disks with handlebody fibers. We give a characterization of fibered homotopy-ribbon disks and give analogues of the Stallings twist for fibered disks and 2{knots. As an application, we produce infinite families of distinct homotopy-ribbon disks with homotopy equivalent exteriors, with potential relevance to the Slice-Ribbon Conjecture. We show that any fibered ribbon 2{ knot can be obtained by doubling infinitely many different slice disks (sometimes in different contractible 4{manifolds). Finally, we illustrate these ideas for the examples arising from spinning fibered 1{knots. 1. Introduction A knot K is fibered if its complement fibers over the circle (with the fibration well-behaved near K). Fibered knots have a long and rich history of study (for both classical knots and higher-dimensional knots). In the classical case, a theorem of Stallings ([Sta62], see also [Neu63]) states that a knot is fibered if and only if its group has a finitely generated commutator subgroup. Stallings [Sta78] also gave a method to produce new fibered knots from old ones by twisting along a fiber, and Harer [Har82] showed that this twisting operation and a type of plumbing is sufficient to generate all fibered knots in S3. Another special class of knots are slice knots. A knot K in S3 is slice if it bounds a smoothly embedded disk in B4 (and more generally an n{knot in Sn+2 is slice if it bounds a disk in Bn+3).
    [Show full text]
  • Categorified Invariants and the Braid Group
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 143, Number 7, July 2015, Pages 2801–2814 S 0002-9939(2015)12482-3 Article electronically published on February 26, 2015 CATEGORIFIED INVARIANTS AND THE BRAID GROUP JOHN A. BALDWIN AND J. ELISENDA GRIGSBY (Communicated by Daniel Ruberman) Abstract. We investigate two “categorified” braid conjugacy class invariants, one coming from Khovanov homology and the other from Heegaard Floer ho- mology. We prove that each yields a solution to the word problem but not the conjugacy problem in the braid group. In particular, our proof in the Khovanov case is completely combinatorial. 1. Introduction Recall that the n-strand braid group Bn admits the presentation σiσj = σj σi if |i − j|≥2, Bn = σ1,...,σn−1 , σiσj σi = σjσiσj if |i − j| =1 where σi corresponds to a positive half twist between the ith and (i + 1)st strands. Given a word w in the generators σ1,...,σn−1 and their inverses, we will denote by σ(w) the corresponding braid in Bn. Also, we will write σ ∼ σ if σ and σ are conjugate elements of Bn. As with any group described in terms of generators and relations, it is natural to look for combinatorial solutions to the word and conjugacy problems for the braid group: (1) Word problem: Given words w, w as above, is σ(w)=σ(w)? (2) Conjugacy problem: Given words w, w as above, is σ(w) ∼ σ(w)? The fastest known algorithms for solving Problems (1) and (2) exploit the Gar- side structure(s) of the braid group (cf.
    [Show full text]
  • Knot Masters Troop 90
    Knot Masters Troop 90 1. Every Scout and Scouter joining Knot Masters will be given a test by a Knot Master and will be assigned the appropriate starting rank and rope. Ropes shall be worn on the left side of scout belt secured with an appropriate Knot Master knot. 2. When a Scout or Scouter proves he is ready for advancement by tying all the knots of the next rank as witnessed by a Scout or Scouter of that rank or higher, he shall trade in his old rope for a rope of the color of the next rank. KNOTTER (White Rope) 1. Overhand Knot Perhaps the most basic knot, useful as an end knot, the beginning of many knots, multiple knots make grips along a lifeline. It can be difficult to untie when wet. 2. Loop Knot The loop knot is simply the overhand knot tied on a bight. It has many uses, including isolation of an unreliable portion of rope. 3. Square Knot The square or reef knot is the most common knot for joining two ropes. It is easily tied and untied, and is secure and reliable except when joining ropes of different sizes. 4. Two Half Hitches Two half hitches are often used to join a rope end to a post, spar or ring. 5. Clove Hitch The clove hitch is a simple, convenient and secure method of fastening ropes to an object. 6. Taut-Line Hitch Used by Scouts for adjustable tent guy lines, the taut line hitch can be employed to attach a second rope, reinforcing a failing one 7.
    [Show full text]