DOCSLIB.ORG

Grid Homology for Knots and Links

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Grid Homology for Knots and Links Grid Homology for Knots and Links Peter S. Ozsvath, Andras I. Stipsicz, and Zoltan Szabo This is a preliminary version of the book Grid Homology for Knots and Links published by the American Mathematical Society (AMS). This preliminary version is made available with the permission of the AMS and may not be changed, edited, or reposted at any other website without explicit written permission from the author and the AMS. Contents Chapter 1. Introduction 1 1.1. Grid homology and the Alexander polynomial 1 1.2. Applications of grid homology 3 1.3. Knot Floer homology 5 1.4. Comparison with Khovanov homology 7 1.5. On notational conventions 7 1.6. Necessary background 9 1.7. The organization of this book 9 1.8. Acknowledgements 11 Chapter 2. Knots and links in S3 13 2.1. Knots and links 13 2.2. Seifert surfaces 20 2.3. Signature and the unknotting number 21 2.4. The Alexander polynomial 25 2.5. Further constructions of knots and links 30 2.6. The slice genus 32 2.7. The Goeritz matrix and the signature 37 Chapter 3. Grid diagrams 43 3.1. Planar grid diagrams 43 3.2. Toroidal grid diagrams 49 3.3. Grids and the Alexander polynomial 51 3.4. Grid diagrams and Seifert surfaces 56 3.5. Grid diagrams and the fundamental group 63 Chapter 4. Grid homology 65 4.1. Grid states 65 4.2. Rectangles connecting grid states 66 4.3. The bigrading on grid states 68 4.4. The simplest version of grid homology 72 4.5. Background on chain complexes 73 4.6. The grid chain complex GC− 75 4.7. The Alexander grading as a winding number 82 4.8. Computations 86 4.9. Further remarks 90 Chapter 5. The invariance of grid homology 91 5.1. Commutation invariance 91 5.2. Stabilization invariance 99 vii viii CONTENTS 5.3. Completion of the invariance proof for grid homology 107 5.4. The destabilization maps, revisited 108 5.5. Other variants of the grid complex 110 5.6. On the holomorphic theory 110 5.7. Further remarks on stabilization maps 110 Chapter 6. The unknotting number and τ 113 6.1. The definition of τ and its unknotting estimate 113 6.2. Construction of the crossing change maps 115 6.3. The Milnor conjecture for torus knots 120 6.4. Canonical grid cycles and estimates on τ 122 Chapter 7. Basic properties of grid homology 127 7.1. Symmetries of the simply blocked grid homology 127 7.2. Genus bounds 129 7.3. General properties of unblocked grid homology 130 7.4. Symmetries of the unblocked theory 132 Chapter 8. The slice genus and τ 135 8.1. Slice genus bounds from τ and their consequences 135 8.2. A version of grid homology for links 136 8.3. Grid homology and saddle moves 139 8.4. Adding unknots to a link 143 8.5. Assembling the pieces: τ bounds the slice genus 146 8.6. The existence of an exotic structure on R4 147 8.7. Slice bounds vs. unknotting bounds 149 Chapter 9. The oriented skein exact sequence 151 9.1. The skein exact sequence 151 9.2. The skein relation on the chain level 153 9.3. Proofs of the skein exact sequences 160 9.4. First computations using the skein sequence 162 9.5. Knots with identical grid homologies 163 9.6. The skein exact sequence and the crossing change map 164 9.7. Further remarks 166 Chapter 10. Grid homologies of alternating knots 167 10.1. Properties of the determinant of a link 167 10.2. The unoriented skein exact sequence 176 10.3. Grid homology groups for alternating knots 183 10.4. Further remarks 185 Chapter 11. Grid homology for links 187 11.1. The definition of grid homology for links 188 11.2. The Alexander multi-grading on grid homology 192 11.3. First examples 194 11.4. Symmetries of grid homology for links 196 11.5. The multi-variable Alexander polynomial 199 11.6. The Euler characteristic of multi-graded grid homology 203 11.7. Seifert genus bounds from grid homology for links 204 CONTENTS ix 11.8. Further examples 205 11.9. Link polytopes and the Thurston norm 210 Chapter 12. Invariants of Legendrian and transverse knots 215 12.1. Legendrian knots in R3 216 12.2. Grid diagrams for Legendrian knots 220 12.3. Legendrian grid invariants 223 12.4. Applications of the Legendrian invariants 228 12.5. Transverse knots in R3 231 12.6. Applications of the transverse invariant 236 12.7. Invariants of Legendrian and transverse links 240 12.8. Transverse knots, grid diagrams, and braids 244 12.9. Further remarks 245 Chapter 13. The filtered grid complex 247 13.1. Some algebraic background 247 13.2. Defining the invariant 252 13.3. Topological invariance of the filtered quasi-isomorphism type 254 13.4. Filtered homotopy equivalences 268 Chapter 14. More on the filtered chain complex 273 14.1. Information in the filtered grid complex 273 14.2. Examples of filtered grid complexes 278 14.3. Refining the Legendrian and transverse invariants: definitions 281 14.4. Applications of the refined Legendrian and transverse invariants 285 14.5. Filtrations in the case of links 287 14.6. Remarks on three-manifold invariants 289 Chapter 15. Grid homology over the integers 291 15.1. Signs assignments and grid homology over Z 291 15.2. Existence and uniqueness of sign assignments 295 15.3. The invariance of grid homology over Z 302 15.4. Invariance in the filtered theory 308 15.5. Other grid homology constructions over Z 319 15.6. On the τ-invariant 321 15.7. Relations in the spin group 321 15.8. Further remarks 323 Chapter 16. The holomorphic theory 325 16.1. Heegaard diagrams 325 16.2. From Heegaard diagrams to holomorphic curves 327 16.3. Multiple basepoints 333 16.4. Equivalence of knot Floer homology with grid homology 335 16.5. Further remarks 338 Chapter 17. Open problems 339 17.1. Open problems in grid homology 339 17.2. Open problems in knot Floer homology 341 Appendix A. Homological algebra 347 A.1. Chain complexes and their homology 347 xCONTENTS A.2. Exact sequences 350 A.3. Mapping cones 352 A.4. On the structure of homology 357 A.5. Dual complexes 359 A.6. On filtered complexes 362 A.7. Small models for filtered grid complexes 363 A.8. Filtered quasi-isomorphism versus filtered homotopy type 365 Appendix B. Basic theorems in knot theory 367 B.1. The Reidemeister Theorem 367 B.2. Reidemeister moves in contact knot theory 373 B.3. The Reidemeister-Singer Theorem 382 B.4. Cromwell’s Theorem 387 B.5. Normal forms of cobordisms between knots 394 Bibliography 399 Index 407 CHAPTER 1 Introduction Heegaard Floer homology is an invariant of closed, oriented three-manifolds, defined by Ozsv´ath and Szab´o by adapting methods from symplectic geometry to Heegaard diagrams [174]. Knot Floer homology is a variation of this construction, discovered in 2003 by Ozsv´ath and Szab´o[172] and independently by Jacob Ras- mussen [191], giving an invariant for knots and links in three-manifolds. It is the homology of a chain complex whose generators can be easily read off from a Hee- gaard diagram, and whose differential counts pseudo-holomorphic disks. Thanks to the contributions of many researchers, the resulting theory has evolved into an active and successful tool for studying knots, links, and three-manifolds. In 2006, Sucharit Sarkar found a method that rendered some of the pseudo- holomorphic disk counts in the Heegaard Floer differential combinatorial; this was further pursued in a joint paper with Jiajun Wang [206]. Inspired by these ideas, Ciprian Manolescu, Ozsv´ath, and Sarkar [135] found a class of Heegaard diagrams for explicitly computing knot Floer homology for knots and links in S3.These Heegaard diagrams are naturally associated to grid diagrams, which are simple combinatorial presentations of knots in S3, dating back to the 19th century [17]. In fact, the above reformulation of knot Floer homology can be used as a def- inition of invariants of knots in S3: their invariance can also be verified using elementary methods, following the paper of Manolescu, Ozsv´ath, Szab´o, and Dylan Thurston [136]. To emphasize this simplicity of the definition, we will call the resulting theory the grid homology for knots and links, to distinguish it from its holomorphic antecedent. Of course, grid homology is isomorphic to knot Floer ho- mology; but owing to its elegance and simplicity, grid homology deserves a purely self-contained treatment. This is the goal of the present book. Before describing the contents of this book in detail, we explain the key features of grid homology. 1.1. Grid homology and the Alexander polynomial Grid homology is defined in terms of a grid presentation of a knot. Such a presentation is given by an n × n grid of squares, n of which are marked with an O and n of which are marked with an X. These are distributed by the rule that each row and each column contains exactly one square marked with an O and exactly one square marked with an X. By connecting the two markings in every row and in every column (orienting the intervals from X to O in the columns and from O to X in the rows) and following the convention that vertical segments pass over horizontal ones, a grid G gives rise to a projection of a knot or link together with an orientation on it.
Load more

Recommended publications
  • Oriented Pair (S 3,S1); Two Knots Are Regarded As
    S-EQUIVALENCE OF KNOTS C. KEARTON Abstract. S-equivalence of classical knots is investigated, as well as its rela- tionship with mutation and the unknotting number. Furthermore, we identify the kernel of Bredon’s double suspension map, and give a geometric relation between slice and algebraically slice knots. Finally, we show that every knot is S-equivalent to a prime knot. 1. Introduction An oriented knot k is a smooth (or PL) oriented pair S3,S1; two knots are regarded as the same if there is an orientation preserving diffeomorphism sending one onto the other. An unoriented knot k is defined in the same way, but without regard to the orientation of S1. Every oriented knot is spanned by an oriented surface, a Seifert surface, and this gives rise to a matrix of linking numbers called a Seifert matrix. Any two Seifert matrices of the same knot are S-equivalent: the definition of S-equivalence is given in, for example, [14, 21, 11]. It is the equivalence relation generated by ambient surgery on a Seifert surface of the knot. In [19], two oriented knots are defined to be S-equivalent if their Seifert matrices are S- equivalent, and the following result is proved. Theorem 1. Two oriented knots are S-equivalent if and only if they are related by a sequence of doubled-delta moves shown in Figure 1. .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .. .... .... .... .... .... .... .... .... .... ...
    [Show full text]
  • Arxiv:Math/0307077V4
    Table of Contents for the Handbook of Knot Theory William W. Menasco and Morwen B. Thistlethwaite, Editors (1) Colin Adams, Hyperbolic Knots (2) Joan S. Birman and Tara Brendle, Braids: A Survey (3) John Etnyre Legendrian and Transversal Knots (4) Greg Friedman, Knot Spinning (5) Jim Hoste, The Enumeration and Classification of Knots and Links (6) Louis Kauffman, Knot Diagramitics (7) Charles Livingston, A Survey of Classical Knot Concordance (8) Lee Rudolph, Knot Theory of Complex Plane Curves (9) Marty Scharlemann, Thin Position in the Theory of Classical Knots (10) Jeff Weeks, Computation of Hyperbolic Structures in Knot Theory arXiv:math/0307077v4 [math.GT] 26 Nov 2004 A SURVEY OF CLASSICAL KNOT CONCORDANCE CHARLES LIVINGSTON In 1926 Artin [3] described the construction of certain knotted 2–spheres in R4. The intersection of each of these knots with the standard R3 ⊂ R4 is a nontrivial knot in R3. Thus a natural problem is to identify which knots can occur as such slices of knotted 2–spheres. Initially it seemed possible that every knot is such a slice knot and it wasn’t until the early 1960s that Murasugi [86] and Fox and Milnor [24, 25] succeeded at proving that some knots are not slice. Slice knots can be used to define an equivalence relation on the set of knots in S3: knots K and J are equivalent if K# − J is slice. With this equivalence the set of knots becomes a group, the concordance group of knots. Much progress has been made in studying slice knots and the concordance group, yet some of the most easily asked questions remain untouched.
    [Show full text]
  • View Given in Figure 7
    New York Journal of Mathematics New York J. Math. 26 (2020) 149{183. Arithmeticity and hidden symmetries of fully augmented pretzel link complements Jeffrey S. Meyer, Christian Millichap and Rolland Trapp Abstract. This paper examines number theoretic and topological prop- erties of fully augmented pretzel link complements. In particular, we determine exactly when these link complements are arithmetic and ex- actly which are commensurable with one another. We show these link complements realize infinitely many CM-fields as invariant trace fields, which we explicitly compute. Further, we construct two infinite fami- lies of non-arithmetic fully augmented link complements: one that has no hidden symmetries and the other where the number of hidden sym- metries grows linearly with volume. This second family realizes the maximal growth rate for the number of hidden symmetries relative to volume for non-arithmetic hyperbolic 3-manifolds. Our work requires a careful analysis of the geometry of these link complements, including their cusp shapes and totally geodesic surfaces inside of these manifolds. Contents 1. Introduction 149 2. Geometric decomposition of fully augmented pretzel links 154 3. Hyperbolic reflection orbifolds 163 4. Cusp and trace fields 164 5. Arithmeticity 166 6. Symmetries and hidden symmetries 169 7. Half-twist partners with many hidden symmetries 174 References 181 1. Introduction 3 Every link L ⊂ S determines a link complement, that is, a non-compact 3 3-manifold M = S n L. If M admits a metric of constant curvature −1, we say that both M and L are hyperbolic, and in fact, if such a hyperbolic Received June 5, 2019.
    [Show full text]
  • Hyperbolic Structures from Link Diagrams
    Hyperbolic Structures from Link Diagrams Anastasiia Tsvietkova, joint work with Morwen Thistlethwaite Louisiana State University/University of Tennessee Anastasiia Tsvietkova, joint work with MorwenHyperbolic Thistlethwaite Structures (UTK) from Link Diagrams 1/23 Every knot can be uniquely decomposed as a knot sum of prime knots (H. Schubert). It also true for non-split links (i.e. if there is no a 2–sphere in the complement separating the link). Of the 14 prime knots up to 7 crossings, 3 are non-hyperbolic. Of the 1,701,935 prime knots up to 16 crossings, 32 are non-hyperbolic. Of the 8,053,378 prime knots with 17 crossings, 30 are non-hyperbolic. Motivation: knots Anastasiia Tsvietkova, joint work with MorwenHyperbolic Thistlethwaite Structures (UTK) from Link Diagrams 2/23 Of the 14 prime knots up to 7 crossings, 3 are non-hyperbolic. Of the 1,701,935 prime knots up to 16 crossings, 32 are non-hyperbolic. Of the 8,053,378 prime knots with 17 crossings, 30 are non-hyperbolic. Motivation: knots Every knot can be uniquely decomposed as a knot sum of prime knots (H. Schubert). It also true for non-split links (i.e. if there is no a 2–sphere in the complement separating the link). Anastasiia Tsvietkova, joint work with MorwenHyperbolic Thistlethwaite Structures (UTK) from Link Diagrams 2/23 Of the 1,701,935 prime knots up to 16 crossings, 32 are non-hyperbolic. Of the 8,053,378 prime knots with 17 crossings, 30 are non-hyperbolic. Motivation: knots Every knot can be uniquely decomposed as a knot sum of prime knots (H.
    [Show full text]
  • A Note on Quasi-Alternating Montesinos Links
    A NOTE ON QUASI-ALTERNATING MONTESINOS LINKS ABHIJIT CHAMPANERKAR AND PHILIP ORDING Abstract. Quasi-alternating links are a generalization of alternating links. They are ho- mologically thin for both Khovanov homology and knot Floer homology. Recent work of Greene and joint work of the first author with Kofman resulted in the classification of quasi- alternating pretzel links in terms of their integer tassel parameters. Replacing tassels by rational tangles generalizes pretzel links to Montesinos links. In this paper we establish conditions on the rational parameters of a Montesinos link to be quasi-alternating. Using recent results on left-orderable groups and Heegaard Floer L-spaces, we also establish con- ditions on the rational parameters of a Montesinos link to be non-quasi-alternating. We discuss examples which are not covered by the above results. 1. Introduction The set Q of quasi-alternating links was defined by Ozsv´athand Szab´o[17] as the smallest set of links satisfying the following: • the unknot is in Q • if link L has a diagram L with a crossing c such that (1) both smoothings of c, L0 and L1 are in Q (2) det(L0) 6= 0 6= det(L1) L L0 L• (3) det(L) = det(L0) + det(L1) then L is in Q. The set Q includes the class of non-split alternating links. Like alternating links, quasi- alternating links are homologically thin for both Khovanov homology and knot Floer ho- mology [14]. The branched double covers of quasi-alternating links are L-spaces [17]. These properties make Q an interesting class to study from the knot homological point of view.
    [Show full text]
  • Some Generalizations of Satoh's Tube
    SOME GENERALIZATIONS OF SATOH'S TUBE MAP BLAKE K. WINTER Abstract. Satoh has defined a map from virtual knots to ribbon surfaces embedded in S4. Herein, we generalize this map to virtual m-links, and use this to construct generalizations of welded and extended welded knots to higher dimensions. This also allows us to construct two new geometric pictures of virtual m-links, including 1-links. 1. Introduction In [13], Satoh defined a map from virtual link diagrams to ribbon surfaces knotted in S4 or R4. This map, which he called T ube, preserves the knot group and the knot quandle (although not the longitude). Furthermore, Satoh showed that if K =∼ K0 as virtual knots, T ube(K) is isotopic to T ube(K0). In fact, we can make a stronger statement: if K and K0 are equivalent under what Satoh called w- equivalence, then T ube(K) is isotopic to T ube(K0) via a stable equivalence of ribbon links without band slides. For virtual links, w-equivalence is the same as welded equivalence, explained below. In addition to virtual links, Satoh also defined virtual arc diagrams and defined the T ube map on such diagrams. On the other hand, in [15, 14], the notion of virtual knots was extended to arbitrary dimensions, building on the geometric description of virtual knots given by Kuperberg, [7]. It is therefore natural to ask whether Satoh's idea for the T ube map can be generalized into higher dimensions, as well as inquiring into a generalization of the virtual arc diagrams to higher dimensions.
    [Show full text]
  • On Spectral Sequences from Khovanov Homology 11
    ON SPECTRAL SEQUENCES FROM KHOVANOV HOMOLOGY ANDREW LOBB RAPHAEL ZENTNER Abstract. There are a number of homological knot invariants, each satis- fying an unoriented skein exact sequence, which can be realized as the limit page of a spectral sequence starting at a version of the Khovanov chain com- plex. Compositions of elementary 1-handle movie moves induce a morphism of spectral sequences. These morphisms remain unexploited in the literature, perhaps because there is still an open question concerning the naturality of maps induced by general movies. In this paper we focus on the spectral sequences due to Kronheimer-Mrowka from Khovanov homology to instanton knot Floer homology, and on that due to Ozsv´ath-Szab´oto the Heegaard-Floer homology of the branched double cover. For example, we use the 1-handle morphisms to give new information about the filtrations on the instanton knot Floer homology of the (4; 5)-torus knot, determining these up to an ambiguity in a pair of degrees; to deter- mine the Ozsv´ath-Szab´ospectral sequence for an infinite class of prime knots; and to show that higher differentials of both the Kronheimer-Mrowka and the Ozsv´ath-Szab´ospectral sequences necessarily lower the delta grading for all pretzel knots. 1. Introduction Recent work in the area of the 3-manifold invariants called knot homologies has il- luminated the relationship between Floer-theoretic knot homologies and `quantum' knot homologies. The relationships observed take the form of spectral sequences starting with a quantum invariant and abutting to a Floer invariant. A primary ex- ample is due to Ozsv´athand Szab´o[15] in which a spectral sequence is constructed from Khovanov homology of a knot (with Z=2 coefficients) to the Heegaard-Floer homology of the 3-manifold obtained as double branched cover over the knot.
    [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]
  • THE JONES SLOPES of a KNOT Contents 1. Introduction 1 1.1. The
    THE JONES SLOPES OF A KNOT STAVROS GAROUFALIDIS Abstract. The paper introduces the Slope Conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural number) of a knot in 3-space. We formulate a number of conjectures for these invariants and verify them by explicit computations for the class of alternating knots, the knots with at most 9 crossings, the torus knots and the (−2, 3,n) pretzel knots. Contents 1. Introduction 1 1.1. The degree of the Jones polynomial and incompressible surfaces 1 1.2. The degree of the colored Jones function is a quadratic quasi-polynomial 3 1.3. q-holonomic functions and quadratic quasi-polynomials 3 1.4. The Jones slopes and the Jones period of a knot 4 1.5. The symmetrized Jones slopes and the signature of a knot 5 1.6. Plan of the proof 7 2. Future directions 7 3. The Jones slopes and the Jones period of an alternating knot 8 4. Computing the Jones slopes and the Jones period of a knot 10 4.1. Some lemmas on quasi-polynomials 10 4.2. Computing the colored Jones function of a knot 11 4.3. Guessing the colored Jones function of a knot 11 4.4. A summary of non-alternating knots 12 4.5. The 8-crossing non-alternating knots 13 4.6.
    [Show full text]
  • Arxiv:2003.00512V2 [Math.GT] 19 Jun 2020 the Vertical Double of a Virtual Link, As Well As the More General Notion of a Stack for a Virtual Link
    A GEOMETRIC INVARIANT OF VIRTUAL n-LINKS BLAKE K. WINTER Abstract. For a virtual n-link K, we define a new virtual link VD(K), which is invariant under virtual equivalence of K. The Dehn space of VD(K), which we denote by DD(K), therefore has a homotopy type which is an invariant of K. We show that the quandle and even the fundamental group of this space are able to detect the virtual trefoil. We also consider applications to higher-dimensional virtual links. 1. Introduction Virtual links were introduced by Kauffman in [8], as a generalization of classical knot theory. They were given a geometric interpretation by Kuper- berg in [10]. Building on Kuperberg's ideas, virtual n-links were defined in [16]. Here, we define some new invariants for virtual links, and demonstrate their power by using them to distinguish the virtual trefoil from the unknot, which cannot be done using the ordinary group or quandle. Although these knots can be distinguished by means of other invariants, such as biquandles, there are two advantages to the approach used herein: first, our approach to defining these invariants is purely geometric, whereas the biquandle is en- tirely combinatorial. Being geometric, these invariants are easy to generalize to higher dimensions. Second, biquandles are rather complicated algebraic objects. Our invariants are spaces up to homotopy type, with their ordinary quandles and fundamental groups. These are somewhat simpler to work with than biquandles. The paper is organized as follows. We will briefly review the notion of a virtual link, then the notion of the Dehn space of a virtual link.
    [Show full text]
  • Arxiv:1501.00726V2 [Math.GT] 19 Sep 2016 3 2 for Each N, Ln Is Assembled from a Tangle S in B , N Copies of a Tangle T in S × I, and the Mirror Image S of S
    HIDDEN SYMMETRIES VIA HIDDEN EXTENSIONS ERIC CHESEBRO AND JASON DEBLOIS Abstract. This paper introduces a new approach to finding knots and links with hidden symmetries using \hidden extensions", a class of hidden symme- tries defined here. We exhibit a family of tangle complements in the ball whose boundaries have symmetries with hidden extensions, then we further extend these to hidden symmetries of some hyperbolic link complements. A hidden symmetry of a manifold M is a homeomorphism of finite-degree covers of M that does not descend to an automorphism of M. By deep work of Mar- gulis, hidden symmetries characterize the arithmetic manifolds among all locally symmetric ones: a locally symmetric manifold is arithmetic if and only if it has infinitely many \non-equivalent" hidden symmetries (see [13, Ch. 6]; cf. [9]). Among hyperbolic knot complements in S3 only that of the figure-eight is arith- metic [10], and the only other knot complements known to possess hidden sym- metries are the two \dodecahedral knots" constructed by Aitchison{Rubinstein [1]. Whether there exist others has been an open question for over two decades [9, Question 1]. Its answer has important consequences for commensurability classes of knot complements, see [11] and [2]. The partial answers that we know are all negative. Aside from the figure-eight, there are no knots with hidden symmetries with at most fifteen crossings [6] and no two-bridge knots with hidden symmetries [11]. Macasieb{Mattman showed that no hyperbolic (−2; 3; n) pretzel knot, n 2 Z, has hidden symmetries [8]. Hoffman showed the dodecahedral knots are commensurable with no others [7].
    [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]