Framed Links Research Articles

From colored triangulations to framed link presentations of 3-manifolds by a polynomial algorithm

For a large class of closed orientable 3-manifolds, we present a polynomial algorithm to go from a triangulation to a framed link presenting the same manifold. The algorithm applies whenever we can get a resoluble gem from the triangulation. Most minimal gems have this property (about 95% of gems in our complete catalogue up to 28 vertices). From the gems which fail, nothing prevents another gem from inducing the same 3-manifold to be resoluble. We leave as an open problem presenting a 3-manifold which is not induced by a resoluble gem. This paper is an application of the theory of gems to 3-manifold topology.

Relative Reshetikhin–Turaev Invariants, Hyperbolic Cone Metrics and Discrete Fourier Transforms I

We propose the Volume Conjecture for the relative Reshetikhin–Turaev invariants of a closed oriented 3-manifold with a colored framed link inside it whose asymptotic behavior is related to the volume and the Chern–Simons invariant of the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring. We prove the conjecture in the case that the ambient 3-manifold is obtained by doing an integral surgery along some components of a fundamental shadow link and the complement of the link in the ambient manifold is homeomorphic to the fundamental shadow link complement, for sufficiently small cone angles. Together with Costantino and Thurston’s result that all compact oriented 3-manifolds with toroidal or empty boundary can be obtained by doing an integral surgery along some components of a suitable fundamental shadow link, this provides a possible approach of solving Chen–Yang’s Volume Conjecture for the Reshetikhin–Turaev invariants of closed oriented hyperbolic 3-manifolds. We also introduce a family of topological operations (the change-of-pair operations) that connect all pairs of a closed oriented 3-manifold and a framed link inside it that have homeomorphic complements, which correspond to doing the partial discrete Fourier transforms to the corresponding relative Reshetikhin–Turaev invariants. As an application, we find a Poisson Summation Formula for the discrete Fourier transforms.

3-Lie algebras, ternary Nambu-Lie algebras and the Yang-Baxter equation

We construct ternary self-distributive (TSD) objects from compositions of binary Lie algebras, 3-Lie algebras and, in particular, ternary Nambu-Lie algebras. We show that the structures obtained satisfy an invertibility property resembling that of racks. We prove that these structures give rise to Yang-Baxter operators in the tensor product of the base vector space and, upon defining suitable twisting isomorphisms, we obtain representations of the infinite (framed) braid group. We consider examples for low-dimensional Lie algebras, where the ternary bracket is defined by composition of the binary ones, along with simple 3-Lie algebras. We show that the Yang-Baxter operators obtained are not gauge equivalent to the transposition operator, and we consider the problem of deforming the operators to obtain new solutions to the Yang-Baxter equation. We discuss the applications of this deformation procedure to the construction of (framed) link invariants.

Kirby diagrams and 5-colored graphs representing compact 4-manifolds

AbstractIt is well-known that in dimension 4 any framed link (L, c) uniquely represents the PL 4-manifold $$M^4(L,c)$$ M 4 ( L , c ) obtained from $${\mathbb {D}}^4$$ D 4 by adding 2-handles along (L, c). Moreover, if trivial dotted components are also allowed (i.e. in case of a Kirby diagram$$(L^{(*)},d)$$ ( L ( ∗ ) , d ) ), the associated PL 4-manifold $$M^4(L^{(*)},d)$$ M 4 ( L ( ∗ ) , d ) is obtained from $${\mathbb {D}}^4$$ D 4 by adding 1-handles along the dotted components and 2-handles along the framed components. In this paper we study the relationships between framed links and/or Kirby diagrams and the representation theory of compact PL manifolds by edge-colored graphs: in particular, we describe how to construct algorithmically a (regular) 5-colored graph representing $$M^4(L^{(*)},d)$$ M 4 ( L ( ∗ ) , d ) , directly “drawn over” a planar diagram of $$(L^{(*)},d)$$ ( L ( ∗ ) , d ) , or equivalently how to algorithmically obtain a triangulation of $$M^4(L^{(*)},d)$$ M 4 ( L ( ∗ ) , d ) . As a consequence, the procedure yields triangulations for any closed (simply-connected) PL 4-manifold admitting handle decompositions without 3-handles. Furthermore, upper bounds for both the invariants gem-complexity and regular genus of $$M^4(L^{(*)},d)$$ M 4 ( L ( ∗ ) , d ) are obtained, in terms of the combinatorial properties of the Kirby diagram.

Spaces of knots in the solid torus, knots in the thickened torus, and links in the 3-sphere

We recursively determine the homotopy type of the space of any irreducible framed link in the 3-sphere, modulo rotations. This leads us to the homotopy type of the space of any knot in the solid torus, thus answering a question posed by Arnold. We similarly study spaces of unframed links in the 3-sphere, modulo rotations, and spaces of knots in the thickened torus. The subgroup of meridional rotations splits as a direct factor of the fundamental group of the space of any framed link except the unknot. Its generators can be viewed as generalizations of the Gramain loop in the space of long knots. Taking the quotient by certain such rotations relates the spaces we study. All of our results generalize previous work of Hatcher and Budney. We provide many examples and explicitly describe generators of fundamental groups.

Linking between singular locus and regular fibers

Given a null-cobordant oriented framed link $L$ in a closed oriented $3$--manifold $M$, we determine those links in $M \setminus L$ which can be realized as the singular point set of a generic map $M \to \mathbb{R}^2$ that has $L$ as an oriented framed regular fiber. Then, we study the linking behavior between the singular point set and regular fibers for generic maps of $M$ into $\mathbb{R}^2$.

Kirby calculus for null-homologous framed links in 3-manifolds

A theorem of Kirby gives a necessary and sufficient condition for two framed links in S^3 to yield orientation-preserving diffeomorphic results of surgery. Kirby's theorem is an important method for constructing invariants of 3-manifolds. In this paper, we prove a variant of Kirby's theorem for null-homologous framed links in a 3-manifold. This result involves a new kind of moves, called IHX-moves, which are closely related to the IHX relation in the theory of finite type invariants. When the first homology group of M is free abelian, we give a refinement of this result to \pm1-framed, algebraically split, null-homologous framed links in M.

The Witten–Reshetikhin–Turaev invariant for links in finite order mapping tori I

We state Asymptotic Expansion and Growth Rate conjectures for the Witten–Reshetikhin–Turaev invariants of arbitrary framed links in 3-manifolds, and we prove these conjectures for the natural links in mapping tori of finite-order automorphisms of marked surfaces. Our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface, which we show coincides with the construction of the Witten–Reshetikhin–Turaev invariants using conformal field theory, as was recently completed by Andersen and Ueno.

On Kirby calculus for null-homotopic framed links in 3–manifolds

Kirby proved that two framed links in S^3 give orientation-preserving homeomorphic results of surgery if and only if these two links are related by a sequence of two kinds of moves called stabilizations and handle-slides. Fenn and Rourke gave a necessary and sufficient condition for two framed links in a closed, oriented 3-manifold to be related by a finite sequence of these moves. The purpose of this paper is twofold. We first give a generalization of Fenn and Rourke's result to 3-manifolds with boundary. Then we apply this result to the case of framed links whose components are null-homotopic in the 3-manifold.

On simplicial resolutions of framed links

In this paper, we investigate the simplicial groups obtained from the link groups of naive cablings on any given framed link. Our main result states that the resulting simplicial groups have the homotopy type of the loop space of a wedge of 3 3 -spheres. This gives simplicial group models for some loop spaces using link groups.

Knowledge-based Links for Automatic Interaction with Programming Online Judges

Programming online judges are computing resources with pre-designed test data. Recently, an increasing attention has been paid on integration of online judges into a tutoring system. However, it is difficult for a local system to automatic interaction with remote online judges on the web to share their computation, because these computing resources are designed for human users only. To address such issue, a novel link is proposed here, called framed link, which consists of a frame-based representation for the knowledge about how to interact with the computing resource that the link points to. With the framed link, three remote online judges including their internal test data are integrated successfully into a local system for programming courses.

Formula omitted]-adic framed braids II

The Yokonuma–Hecke algebras are quotients of the modular framed braid group and they support Markov traces. In this paper, which is sequel to Juyumaya and Lambropoulou (2007) [6], we explore further the structures of the p-adic framed braids and the p-adic Yokonuma–Hecke algebras constructed by Juyumaya and Lambropoulou (2007) [6], by means of dense sub-structures approximating p-adic elements. We also construct a p-adic Markov trace on the p-adic Yokonuma–Hecke algebras and approximate the values of the p-adic trace on p-adic elements. Surprisingly, the Markov traces do not re-scale directly to yield isotopy invariants of framed links. This leads to imposing the ‘E-condition’ on the trace parameters. For solutions of the ‘E-system’ we then define 2-variable isotopy invariants of modular framed links. These lift to p-adic isotopy invariants of classical framed links. The Yokonuma–Hecke algebras have topological interpretations in the context of framed knots, of classical knots of singular knots and of transverse knots.

Artin presentations and fundamental groups of 3-manifolds

We prove that the set of n-Artin presentations has a group structure. It is a known result but it seems that it does not appear explicitly in the literature. As an application we consider a special class of integral framed links β ˆ in S 3 such that β = ∏ i = 1 n Δ 2 e σ 1 2 e i σ 2 2 f i and we calculate the fundamental group of the 3-manifolds obtained by integral Dehn surgery on these links. In some cases we say when the groups obtained cannot be trivial.

On the Naturality of the Spectral Sequence from Khovanov Homology to Heegaard Floer Homology

In [18], Ozsvath-Szabo established an algebraic relationship, in the form of a spectral sequence, between the reduced Khovanov homology of (the mirror of) a link and the Heegaard Floer homology of its double-branched cover. This relationship, extended in [19] and [4], was recast, in [5], as a specific instance of a broader connection between Khovanov- and Heegaard Floer-type homology theories, using a version of Heegaard Floer homology for sutured manifolds developed by Juhasz in [7]. In the present work, we prove the naturality of the spectral sequence under certain elementary operations, using a generalization of Juhasz’s surface decomposition theorem valid for decomposing surfaces geometrically disjoint from an imbedded framed link.

Goussarov–Habiro theory for string links and the Milnor–Johnson correspondence

We study the Goussarov–Habiro finite type invariants theory for framed string links in homology balls. Their degree 1 invariants are computed: they are given by Milnor's triple linking numbers, the mod 2 reduction of the Sato–Levine invariant, Arf and Rochlin's μ invariant. These invariants are seen to be naturally related to invariants of homology cylinders through the Milnor–Johnson correspondence: in particular, an analogue of the Birman–Craggs homomorphism for string links is computed. The relation with Vassiliev theory is studied.

TOPOLOGICAL HELICITY FOR FRAMED LINKS

We introduce topological helicity, an invariant for oriented framed links. Topological helicity provides an elementary means of computing helicity for a magnetic flux rope by measuring its knotting, linking, and twisting. We present an equivalence relation, reconnection-equivalence, for framed links and prove that topological helicity is a complete invariant for the resulting equivalence classes. We conclude by showing that one can use magnetic reconnection to transform one collection of linked flux ropes into another collection if and only if they have the same helicity.

Dotted Links, Heegaard Diagrams, and Colored Graphs for PL 4-manifolds

The present paper is devoted to establish a connection between the 4-manifold representation method by dotted framed links (or—in the closed case—by Heegaard diagrams) and the so called crystallization theory, which visualizes general PL-manifolds by means of edge-colored graphs. In particular, it is possible to obtain a crystallization of a closed 4-manifold M4 starting from a Heegaard diagram (#m(S1×S2), !), and the algorithmicity of the whole process depends on the effective possibility of recognizing (#m(S1×S2), !) to be a Heegaard diagram by crystallization theory.

The fourth Skein Module and the Montesinos-Nakanishi Conjecture for 3-algebraic links

We study the fourth skein module of 3-manifolds, based on the skein relation b0L0 + b1L1 + b2L2 + b3L3 = 0 and a framing relation L(1) = aL (a, b0, b3 invertible). We give necessary conditions for trivial links to be linearly independent in the module. We show how elements of the skein module behave under the n-move and we compute the values for (2, n)-torus links and twist knots as elements of the skein module. Using mutants and rotors, we find different links which represent the same element in the skein module. We also show that algebraic links (in the sense of Conway) and closed 3-braids are linear combinations of trivial links. We introduce the concept of n-algebraic tangles (and links) and analyze the skein module for 3-algebraic links. As a by product we prove the Montesinos-Nakanishi 3-moves conjecture for 3-algebraic links (including 3-bridge links). For links in S3, the structure of our skein module suggests the existence of three new polynomial invariants of unoriented framed (or unframed) links. One of them would generalize the Kauffman polynomial of links and another one could be used to analyze amphicheirality of links (and may work better than the Kauffman polynomial). In the conclusion, we speculate that our new knot invariants are related to a deformation of the symplectic quotient of braid groups.

RACKS AND LINKS IN CODIMENSION TWO

A rack, which is the algebraic distillation of two of the Reidemeister moves, is a set with a binary operation such that right multiplication is an automorphism. Any codimension two link has a fundamental rack which contains more information than the fundamental group. Racks provide an elegant and complete algebraic framework in which to study links and knots in 3–manifolds, and also for the 3–manifolds themselves. Racks have been studied by several previous authors and have been called a variety of names. In this first paper of a series we consolidate the algebra of racks and show that the fundamental rack is a complete invariant for irreducible framed links in a 3–manifold and for the 3–manifold itself. We give some examples of computable link invariants derived from the fundamental rack and explain the connection of the theory of racks with that of braids.

THE UNIVERSAL LINK POLYNOMIAL

The solution of the non-Abelian SU (N) quantum Chern–Simons field theory defined in R3 is presented. It is shown how to compute the expectation values of the Wilson line operators, associated with oriented framed links, in closed form. The main properties of the universal link polynomial, defined by these expectation values, are derived in the case of a generic real simple Lie algebra. The resulting polynomials for some simple examples of links are reported.