Quantum Representations and Monodromies of Fibered Links
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Finite Type Invariants for Knots 3-Manifolds
Pergamon Topology Vol. 37, No. 3. PP. 673-707, 1998 ~2 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0040-9383/97519.00 + 0.00 PII: soo4o-9383(97)00034-7 FINITE TYPE INVARIANTS FOR KNOTS IN 3-MANIFOLDS EFSTRATIA KALFAGIANNI + (Received 5 November 1993; in revised form 4 October 1995; final version 16 February 1997) We use the Jaco-Shalen and Johannson theory of the characteristic submanifold and the Torus theorem (Gabai, Casson-Jungreis)_ to develop an intrinsic finite tvne__ theory for knots in irreducible 3-manifolds. We also establish a relation between finite type knot invariants in 3-manifolds and these in R3. As an application we obtain the existence of non-trivial finite type invariants for knots in irreducible 3-manifolds. 0 1997 Elsevier Science Ltd. All rights reserved 0. INTRODUCTION The theory of quantum groups gives a systematic way of producing families of polynomial invariants, for knots and links in [w3 or S3 (see for example [18,24]). In particular, the Jones polynomial [12] and its generalizations [6,13], can be obtained that way. All these Jones-type invariants are defined as state models on a knot diagram or as traces of a braid group representation. On the other hand Vassiliev [25,26], introduced vast families of numerical knot invariants (Jinite type invariants), by studying the topology of the space of knots in [w3. The compu- tation of these invariants, involves in an essential way the computation of related invariants for special knotted graphs (singular knots). It is known [l-3], that after a suitable change of variable the coefficients of the power series expansions of the Jones-type invariants, are of Jinite type. -
Part III 3-Manifolds Lecture Notes C Sarah Rasmussen, 2019
Part III 3-manifolds Lecture Notes c Sarah Rasmussen, 2019 Contents Lecture 0 (not lectured): Preliminaries2 Lecture 1: Why not ≥ 5?9 Lecture 2: Why 3-manifolds? + Intro to Knots and Embeddings 13 Lecture 3: Link Diagrams and Alexander Polynomial Skein Relations 17 Lecture 4: Handle Decompositions from Morse critical points 20 Lecture 5: Handles as Cells; Handle-bodies and Heegard splittings 24 Lecture 6: Handle-bodies and Heegaard Diagrams 28 Lecture 7: Fundamental group presentations from handles and Heegaard Diagrams 36 Lecture 8: Alexander Polynomials from Fundamental Groups 39 Lecture 9: Fox Calculus 43 Lecture 10: Dehn presentations and Kauffman states 48 Lecture 11: Mapping tori and Mapping Class Groups 54 Lecture 12: Nielsen-Thurston Classification for Mapping class groups 58 Lecture 13: Dehn filling 61 Lecture 14: Dehn Surgery 64 Lecture 15: 3-manifolds from Dehn Surgery 68 Lecture 16: Seifert fibered spaces 69 Lecture 17: Hyperbolic 3-manifolds and Mostow Rigidity 70 Lecture 18: Dehn's Lemma and Essential/Incompressible Surfaces 71 Lecture 19: Sphere Decompositions and Knot Connected Sum 74 Lecture 20: JSJ Decomposition, Geometrization, Splice Maps, and Satellites 78 Lecture 21: Turaev torsion and Alexander polynomial of unions 81 Lecture 22: Foliations 84 Lecture 23: The Thurston Norm 88 Lecture 24: Taut foliations on Seifert fibered spaces 89 References 92 1 2 Lecture 0 (not lectured): Preliminaries 0. Notation and conventions. Notation. @X { (the manifold given by) the boundary of X, for X a manifold with boundary. th @iX { the i connected component of @X. ν(X) { a tubular (or collared) neighborhood of X in Y , for an embedding X ⊂ Y . -
Ordering Thurston's Geometries by Maps of Non-Zero Degree
ORDERING THURSTON’S GEOMETRIES BY MAPS OF NON-ZERO DEGREE CHRISTOFOROS NEOFYTIDIS ABSTRACT. We obtain an ordering of closed aspherical 4-manifolds that carry a non-hyperbolic Thurston geometry. As application, we derive that the Kodaira dimension of geometric 4-manifolds is monotone with respect to the existence of maps of non-zero degree. 1. INTRODUCTION The existence of a map of non-zero degree defines a transitive relation, called domination rela- tion, on the homotopy types of closed oriented manifolds of the same dimension. Whenever there is a map of non-zero degree M −! N we say that M dominates N and write M ≥ N. In general, the domain of a map of non-zero degree is a more complicated manifold than the target. Gromov suggested studying the domination relation as defining an ordering of compact oriented manifolds of a given dimension; see [3, pg. 1]. In dimension two, this relation is a total order given by the genus. Namely, a surface of genus g dominates another surface of genus h if and only if g ≥ h. However, the domination relation is not generally an order in higher dimensions, e.g. 3 S3 and RP dominate each other but are not homotopy equivalent. Nevertheless, it can be shown that the domination relation is a partial order in certain cases. For instance, 1-domination defines a partial order on the set of closed Hopfian aspherical manifolds of a given dimension (see [18] for 3- manifolds). Other special cases have been studied by several authors; see for example [3, 1, 2, 25]. -
Computing Triangulations of Mapping Tori of Surface Homeomorphisms
Computing Triangulations of Mapping Tori of Surface Homeomorphisms Peter Brinkmann∗ and Saul Schleimer Abstract We present the mathematical background of a software package that computes triangulations of mapping tori of surface homeomor- phisms, suitable for Jeff Weeks’s program SnapPea. The package is an extension of the software described in [?]. It consists of two programs. jmt computes triangulations and prints them in a human-readable format. jsnap converts this format into SnapPea’s triangulation file format and may be of independent interest because it allows for quick and easy generation of input for SnapPea. As an application, we ob- tain a new solution to the restricted conjugacy problem in the mapping class group. 1 Introduction In [?], the first author described a software package that provides an en- vironment for computer experiments with automorphisms of surfaces with one puncture. The purpose of this paper is to present the mathematical background of an extension of this package that computes triangulations of mapping tori of such homeomorphisms, suitable for further analysis with Jeff Weeks’s program SnapPea [?].1 ∗This research was partially conducted by the first author for the Clay Mathematics Institute. 2000 Mathematics Subject Classification. 57M27, 37E30. Key words and phrases. Mapping tori of surface automorphisms, pseudo-Anosov au- tomorphisms, mapping class group, conjugacy problem. 1Software available at http://thames.northnet.org/weeks/index/SnapPea.html 1 Pseudo-Anosov homeomorphisms are of particular interest because their mapping tori are hyperbolic 3-manifolds of finite volume [?]. The software described in [?] recognizes pseudo-Anosov homeomorphisms. Combining this with the programs discussed here, we obtain a powerful tool for generating and analyzing large numbers of hyperbolic 3-manifolds. -
Singular Hyperbolic Structures on Pseudo-Anosov Mapping Tori
SINGULAR HYPERBOLIC STRUCTURES ON PSEUDO-ANOSOV MAPPING TORI A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Kenji Kozai June 2013 iv Abstract We study three-manifolds that are constructed as mapping tori of surfaces with pseudo-Anosov monodromy. Such three-manifolds are endowed with natural sin- gular Sol structures coming from the stable and unstable foliations of the pseudo- Anosov homeomorphism. We use Danciger’s half-pipe geometry [6] to extend results of Heusener, Porti, and Suarez [13] and Hodgson [15] to construct singular hyperbolic structures when the monodromy has orientable invariant foliations and its induced action on cohomology does not have 1 as an eigenvalue. We also discuss a combina- torial method for deforming the Sol structure to a singular hyperbolic structure using the veering triangulation construction of Agol [1] when the surface is a punctured torus. v Preface As a result of Perelman’s Geometrization Theorem, every 3-manifold decomposes into pieces so that each piece admits one of eight model geometries. Most 3-manifolds ad- mit hyperbolic structures, and outstanding questions in the field mostly center around these manifolds. Having an effective method for describing the hyperbolic structure would help with understanding topology and geometry in dimension three. The recent resolution of the Virtual Fibering Conjecture also means that every closed, irreducible, atoroidal 3-manifold can be constructed, up to a finite cover, from a homeomorphism of a surface by taking the mapping torus for the homeomorphism [2]. -
Inventiones Mathematicae 9 Springer-Verlag 1994
Invent. math. 118, 255 283 (1994) Inventiones mathematicae 9 Springer-Verlag 1994 Bundles and finite foliations D. Cooper 1'*, D.D. Long 1'**, A.W. Reid 2 .... 1 Department of Mathematics, University of California, Santa Barbara CA 93106, USA z Department of Pure Mathematics, University of Cambridge, Cambridge CB2 1SB, UK Oblatum 15-1X-1993 & 30-X11-1993 1 Introduction By a hyperbolic 3-manifold, we shall always mean a complete orientable hyperbolic 3-manifold of finite volume. We recall that if F is a Kleinian group then it is said to be geometrically finite if there is a finite-sided convex fundamental domain for the action of F on hyperbolic space. Otherwise, s is geometrically infinite. If F happens to be a surface group, then we say it is quasi-Fuchsian if the limit set for the group action is a Jordan curve C and F preserves the components of S~\C. The starting point for this work is the following theorem, which is a combination of theorems due to Marden [10], Thurston [14] and Bonahon [1]. Theorem 1.1 Suppose that M is a closed orientable hyperbolic 3-man!fold. If g: Sq-~ M is a lrl-injective map of a closed surface into M then exactly one of the two alternatives happens: 9 The 9eometrically infinite case: there is a finite cover lVl of M to which g l([ts and can be homotoped to be a homeomorphism onto a fiber of some fibration of over the circle. 9 The 9eometricallyfinite case: g,~zl (S) is a quasi-Fuchsian group. -
Endomorphisms of Mapping Tori
ENDOMORPHISMS OF MAPPING TORI CHRISTOFOROS NEOFYTIDIS Abstract. We classify in terms of Hopf-type properties mapping tori of residually finite Poincar´eDuality groups with non-zero Euler characteristic. This generalises and gives a new proof of the analogous classification for fibered 3-manifolds. Various applications are given. In particular, we deduce that rigidity results for Gromov hyperbolic groups hold for the above mapping tori with trivial center. 1. Introduction We classify in terms of Hopf-type properties mapping tori of residually finite Poincar´eDu- ality (PD) groups K with non-zero Euler characteristic, which satisfy the following finiteness condition: if there exists an integer d > 1 and ξ 2 Aut(K) such that θd = ξθξ−1 mod Inn(K); (∗) then θq 2 Inn(K) for some q > 1: One sample application is that every endomorphism onto a finite index subgroup of the 1 fundamental group of a mapping torus F oh S , where F is a closed aspherical manifold with 1 π1(F ) = K as above, induces a homotopy equivalence (equivalently, π1(F oh S ) is cofinitely 1 1 Hopfian) if and only if the center of π1(F oh S ) is trivial; equivalently, π1(F oh S ) is co- Hopfian. If, in addition, F is topologically rigid, then homotopy equivalence can be replaced by a map homotopic to a homeomorphism, by Waldhausen's rigidity [Wa] in dimension three and Bartels-L¨uck's rigidity [BL] in dimensions greater than four. Condition (∗) is known to be true for every aspherical surface by a theorem of Farb, Lubotzky and Minsky [FLM] on translation lengths. -
On Open Book Embedding of Contact Manifolds in the Standard Contact Sphere
ON OPEN BOOK EMBEDDING OF CONTACT MANIFOLDS IN THE STANDARD CONTACT SPHERE KULDEEP SAHA Abstract. We prove some open book embedding results in the contact category with a constructive ap- proach. As a consequence, we give an alternative proof of a Theorem of Etnyre and Lekili that produces a large class of contact 3-manifolds admitting contact open book embeddings in the standard contact 5-sphere. We also show that all the Ustilovsky (4m + 1)-spheres contact open book embed in the standard contact (4m + 3)-sphere. 1. Introduction An open book decomposition of a manifold M m is a pair (V m−1; φ), such that M m is diffeomorphic to m−1 m−1 2 m−1 MT (V ; φ)[id @V ×D . Here, V , the page, is a manifold with boundary, and φ, the monodromy, is a diffeomorphism of V m−1 that restricts to identity in a neighborhood of the boundary @V . MT (V m−1; φ) denotes the mapping torus of φ. We denote an open book, with page V m−1 and monodromy φ, by Aob(V; φ). The existence of open book decompositions, for a fairly large class of manifolds, is now known due the works of Alexander [Al], Winkelnkemper [Wi], Lawson [La], Quinn [Qu] and Tamura [Ta]. In particular, every closed, orientable, odd dimensional manifold admits an open book decomposition. Thurston and Winkelnkemper [TW] have shown that starting from an exact symplectic manifold (Σ2m;!) 2m as page and a boundary preserving symplectomorphism φs of (Σ ;!) as monodromy, one can produce a 2m+1 2m contact 1-form α on N = Aob(Σ ; φs). -
Topology and Geometry of 2 and 3 Dimensional Manifolds
Topology and Geometry of 2 and 3 dimensional manifolds Chris John May 3, 2016 Supervised by Dr. Tejas Kalelkar 1 Introduction In this project I started with studying the classification of Surface and then I started studying some preliminary topics in 3 dimensional manifolds. 2 Surfaces Definition 2.1. A simple closed curve c in a connected surface S is separating if S n c has two components. It is non-separating otherwise. If c is separating and a component of S n c is an annulus or a disk, then it is called inessential. A curve is essential otherwise. An observation that we can make here is that a curve is non-separating if and only if its complement is connected. For manifolds, connectedness implies path connectedness and hence c is non-separating if and only if there is some other simple closed curve intersecting c exactly once. 2.1 Decomposition of Surfaces Lemma 2.2. A closed surface S is prime if and only if S contains no essential separating simple closed curve. Proof. Consider a closed surface S. Let us assume that S = S1#S2 is a non trivial con- nected sum. The curve along which S1 n(disc) and S2 n(disc) where identified is an essential separating curve in S. Hence, if S contains no essential separating curve, then S is prime. Now assume that there exists a essential separating curve c in S. This means neither of 2 2 the components of S n c are disks i.e. S1 6= S and S2 6= S . -
Meetings & Conferences of The
Meetings & Conferences of the AMS IMPORTANT INFORMATION REGARDING MEETINGS PROGRAMS: AMS Sectional Meeting programs do not appear in the print version of the Notices. However, comprehensive and continually updated meeting and program information with links to the abstract for each talk can be found on the AMS website. See http://www.ams.org/meetings/. Final programs for Sectional Meetings will be archived on the AMS website accessible from the stated URL and in an electronic issue of the Notices as noted below for each meeting. sity, Youssef Raffoul, University of Dayton, and Agacik St. Louis, Missouri Zafer, American University of the Middle East. Advances in Mathematical Methods for Disease Model- Washington University ing, Jimin Ding, Washington University in St. Louis, Necibe October 18–20, 2013 Tuncer, University of Tulsa, and Naveen K. Vaidya, Uni- versity of Missouri-Kansas City. Friday – Sunday Algebraic Cycles and Coherent Sheaves, Roya Beheshti, Meeting #1094 Matt Kerr, and N. Mohan Kumar, Washington University in St. Louis. Central Section Associate secretary: Georgia M. Benkart Algebraic and Combinatorial Invariants of Knots, Announcement issue of Notices: August 2013 Heather Dye, McKendree University, Allison Henrich, Program first available on AMS website: September 5, 2013 Seattle University, Aaron Kaestner, North Park University, Program issue of electronic Notices: October 2013 and Louis Kauffman, University of Illinois. Issue of Abstracts: Volume 34, Issue 4 Automorphic Forms and Representation Theory, Du- bravka Ban and Joe Hundley, Southern Illinois University, Deadlines and Shuichiro Takeda, University of Missouri, Columbia. For organizers: Expired Commutative Algebra, Lianna Sega, University of Mis- For abstracts: Expired souri, Kansas City, and Hema Srinivasan, University of Missouri, Columbia. -
Quantum Products for Mapping Tori and the Atiyah-Floer Conjecture
Quantum products for mapping tori and the Atiyah-Floer conjecture Dietmar A. Salamon ETH-Zuric¨ h 15 July 1999 (revised 14 December 2000) 1 Introduction Let Σ be a compact oriented Riemann surface with area form ! and f0; f1 : : : ; fn be orientation and area preserving diffeomorphisms of Σ such that fn ◦ · · · ◦ f0 = id: Suppose that P ! Σ is a principal SO(3)-bundle with nonzero second Stiefel- Whitney class and choose lifts f~ P −!j P # # f Σ −!j Σ to bundle automorphisms that also satisfy f~n ◦ · · · ◦ f~0 = id. These lifts induce symplectomorphisms φ : M ! M on the moduli space of flat connections f~j Σ Σ on P . They also determine SO(3)-bundles Q ! Y over mapping tori. There f~j fj are natural product structures HF(Y 0 ; Q ) ⊗ HF(Y 1 ; Q ) ⊗ · · · ⊗ HF(Y ; Q ) ! Z (1) f f~0 f f~1 fn f~n in instanton Floer homology and HFsymp(M ; φ ) ⊗ HFsymp(M ; φ ) ⊗ · · · ⊗ HFsymp(M ; φ ) ! Z (2) Σ f~0 Σ f~1 Σ f~n in symplectic Floer homology. Both were discovered by Donaldson. They are well defined up to an overall sign. The homomorphism (1) can be interpreted as a relative Donaldson invariant on a 4-manifold with boundary. In other words, this product is obtained by counting anti-self-dual connections over a 4-dimensional cobordism with n + 1 cylindrical ends corresponding to Yfj . The second homomorphism is obtained by counting pseudo-holomorphic sections of a 1 symplectic fibre bundle over the punctured sphere with fibre MΣ and holonomies φf~j around the n + 1 punctures. -
Crosscap Numbers and the Jones Polynomial
CROSSCAP NUMBERS AND THE JONES POLYNOMIAL EFSTRATIA KALFAGIANNI AND CHRISTINE RUEY SHAN LEE Abstract. We give sharp two-sided linear bounds of the crosscap number (non-orientable genus) of alternating links in terms of their Jones polynomial. Our estimates are often exact and we use them to calculate the crosscap numbers for several infinite families of alternating links and for several alternating knots with up to twelve crossings. We also discuss generalizations of our results for classes of non-alternating links. 1. Introduction The goal of this paper is to give two-sided linear bounds of the crosscap number (i.e. the non-orientable genus) of an alternating link in terms of coefficients of the Jones polynomial of the link. We show that both of these bounds are sharp and often they give the exact value of the crosscap number. As an application we calculate the crosscap number of several infinite families of alternating links. We also check that our bounds give the crosscap numbers of 283 alternating knots with up to twelve crossings that were previously unknown. Finally, we generalize our results to classes of non-alternating links. To state our results, for a link K ⊂ S3, let n n−1 ′ s+1 ′ s JK (t)= αKt + βK t + . + βK t + αKt denote the Jones polynomial of K, so that n and s denote the highest and lowest power in t. Set ′ TK := |βK| + βK , ′ where βK and βK denote the second and penultimate coefficients of JK(t), respectively. Also let sK = n − s, denote the degree span of JK (t).