Determining Knotsw and Links by Cyclic Branched Coverings

It is well known that different knots or links in the 3-sphere can have homeomorphic n-fold cyclic branched coverings. We consider the following problem: for which values of nis a knot of link determined by itsn-fold cyclic branched covering? We consider the class of hyperbolic resp.2π/n-hyperbolic links. The isometry or symmetry groups of such links are finite, and their n-fold branched coverings are hyperbolic 3-manifolds. Our main result states that if ndoes not divide the order of the finite symmetry group of such a link, then the link is determined by its n-fold branched covering. In a sense, the result is best possible; the key argument of its proof is algebraic using some basic result about finite p-groups. The main result applies, for example, to the cyclic branched coverings of the 2-bridge links; in particular, it gives a classification of the maximally symmetricD₆-manifolds which are exactly the 3-fold branched coverings of the 2-bridge links.     

An infinite family of convex Brunnian links in $${\mathbb{R}^n}$$

This paper proves that convex Brunnian links exist for every dimension n ≥ 3 by constructing explicit examples. These examples are three-component links which are higher-dimensional generalizations of the Borromean rings.     

Covering Sato-Levine invariants

Two covering versions of the Sato-Levine invariant are constructed which provide obstructions to certain two-component oriented links in the 3-sphere being link concordant to boundary links. These covering invariants are rational functions one of which detects both nonamphicheirality and noninvertibility of oriented links.

     

Singular Links of Almost Metastable Dimensions

The objects studied are singular links of spheres of dimensions p₁,..., p_r, and p in the n-sphere. A theory of such singular links for the case where max{p₁,..., p_r} < min{2n/3 − 1, n − (p + 5)/3} is constructed. The theory generalizes (as far as it is possible) the theory of singular links of spheres of dimensions k,..., k, and p in the (2k + 1)-sphere, where k > 1, developed in the author's recent papers. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 287–294.     

Singular Links of Two Circles and a Wedge of Circles in the 3-Sphere

A homotopy classification of singular links of two circles and a wedge of circles in the 3-sphere is given. This result generalizes Milnor's one on homotopy classification of classical three-component links. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 295–299.     

Diffeomorphic vs Isotopic Links in Lens Spaces

Links in lens spaces may be defined to be equivalent by ambient isotopy or by diffeomorphism of pairs. In the first case, for all the combinatorial representations of links, there is a set of Reidemeister-type moves on diagrams connecting isotopy equivalent links. In this paper, we provide a set of moves on disk, band and grid diagrams that connects diffeo-equivalent links: there are up to four isotopy equivalent links in each diffeo-equivalence class. Moreover, we investigate how the diffeo-equivalence relates to the lift of the link in the 3-sphere: in the particular case of oriented primitive-homologous knots, the lift completely determines the knot class in L(p, q) up to diffeo-equivalence, and thus only four possible knots up to isotopy equivalence can have the same lift.     

Strongly invertible links and divides

To a proper generic immersion of a finite number of copies of the unit interval in a 2-disc, called a divide, A’Campo associates a link in S³. From the more general notion of ordered Morse signed divides, one obtains a braid presentation of links of divides. In this paper, we prove that every strongly invertible link is isotopic to the link of an ordered Morse signed divide. We give fundamental moves for ordered Morse signed divides and show that strongly invertible links are equivalent if and only if we can pass from one ordered Morse signed divide to the other by a sequence of such moves. Then we associate a polynomial to an ordered Morse signed divide, invariant for these moves. So this polynomial is invariant for the equivalence of strongly invertible links.     

Homotopy Classification of Singular Links of Type (1,1,m;3) with m>1

A homotopy classification of three-component singular links with components of dimensions 1, 1, and m > 1 in the three-sphere S³ is obtained. It is shown that there are links of such type that are pseudo-homotopic but not homotopic. Bibliography: 8 titles.     

On alternation numbers of links

We construct infinitely many hyperbolic links with x-distance far from the set of (possibly, splittable) alternating links in the concordance class of every link. A sensitive result is given for the concordance class of every (possibly, split) alternating link. Our proof uses an estimate of the τ-distance by an Alexander invariant and the topological imitation theory, both established earlier by the author.     

Closed incompressible surfaces of genus two in 3-bridge knot complements

In this paper, we characterize closed incompressible surfaces of genus two in the complements of 3-bridge knots and links. This characterization includes that of essential 2-string tangle decompositions for 3-bridge knots and links.     

On certain classes of hyperbolic 3-manifolds

We study the topological structure and the homeomorphism problem for closed 3-manifolds M(n,k) obtained by pairwise identifications of faces in the boundary of certain polyhedral 3-balls. We prove that they are (n/d)-fold cyclic coverings of the 3-sphere branched over certain hyperbolic links of d+1 components, where d= (n/k). Then we study the closed 3-manifolds obtained by Dehn surgeries on the components of these links. Received: 27 November 1998 / Accepted: 12 May 1999     

Signature invariants of covering links

We apply the theory of signature invariants of links in rational homology spheres to covering links of homology boundary links. From patterns and Seifert matrices of homology boundary links, we derive an explicit formula to compute signature invariants of their covering links. Using the formula, we produce fused boundary links that are positive mutants of ribbon links but are not concordant to boundary links. We also show that for any finite collection of patterns, there are homology boundary links that are not concordant to any homology boundary links admitting a pattern in the collection.

     

Coloured Jones invariants of links and the Volume Conjecture

We extend the definition of the coloured Jones polynomials toframed links and trivalent graphs in S³# k (S² x S¹) using astate-sum formulation based on Turaev's shadows. Then, we provethat the natural extension of the Volume Conjecture holds truefor an infinite family of hyperbolic links.     

Three-page approach to knot theory. Encoding and local moves

In the present paper, we suggest a new combinatorial approach to knot theory based on embeddings of knots and links into a union of three half-planes with the same boundary. The idea to embed knots into a “book” is quite natural and was considered already in [1]. Among recent papers on embeddings of knots into a book with infinitely many pages, we mention [2] and [3] (see also references therein). The restriction of the number of pages to three (or any other number ≥3) provides a convenient way toencode links by words in a finite alphabet. For those words, we give a finite set of local changes that realizes the equivalence of links by analogy with the Reidemeister moves for planar link diagrams. This work is partially supported by Russian Foundation for Basic Research grant No. 99-01-00090. Moscow State University. Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 33, No. 4, pp. 25–37, October–December, 1999. Translated by I. A. Dynnikov     

Khovanov-Jacobsson numbers and invariants of surface-knots derived from Bar-Natan's theory

Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov's theory is functorial for link cobordisms between classical links, we obtain an invariant of a surface-knot, called the Khovanov-Jacobsson number, by considering the surface-knot as a link cobordism between empty links. In this paper, we study an extension of the Khovanov-Jacobsson number derived from Bar-Natan's theory, and prove that any -knot has trivial Khovanov-Jacobsson number.

     

Representing links in 3-manifolds¶ by branched coverings of S3

We introduce a planar coloured-diagram representation of links in 3-manifolds given as branched coverings of the 3-sphere. We also prove an equivalence theorem based on local moves and the existence of a universal configuration for such representation. As an application we give unified proofs of two different results on existence of fibered links in 3-manifolds. Received: 7 April 1997     

Classification of oriented Montesinos links by invariants of spin structures

In this paper we give the isotopy classification of oriented Montesinos links. The definition of the invariants of links needed for this and the proof of the classification theorem are based on a new construction, which establishes a correspondence between orientations of a link S ⁿ on the one hand, and spin structures on the two-sheeted branched covering of the sphere, branched over l, on the other. New numerical invariants of spin structures on three-dimensional Seifert manifolds are introduced in the paper; these invariants are used to classify the Montesinos links.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 143, pp. 130–146, 1985.     

Refined Kirby calculus for three-manifolds of first homology groups of odd prime orders

Every integral homology 3-sphere is presented by a framed link with framing ±1 and without linking numbers. Restricting such presentations, Habiro arranged Kirby calculus so that it preserves framings and linkings and moreover showed that his calculus suffices to relate all links with the same results. This paper provides an extension of his result for manifolds of first homology groups of odd prime orders. After defining our set of links, we establish Habiro calculus over it, and show that, for many orders, it works on those manifolds. We further give the existence of the Casson-Walker invariant for them.     

Efficient automatic simulation of parallel computation on networks of workstations

Andrews et al. [Automatic method for hiding latency in high bandwidth networks, in: Proceedings of the ACM Symposium on Theory of Computing, 1996, pp. 257-265; Improved methods for hiding latency in high bandwidth networks, in: Proceedings of the Eighth Annual ACM Symposium on Parallel Algorithms and Architectures, 1996, pp. 52-61] introduced a number of techniques for automatically hiding latency when performing simulations of networks with unit delay links on networks with arbitrary unequal delay links. In their work, they assume that processors of the host network are identical in computational power to those of the guest network being simulated. They further assume that the links of the host are able to pipeline messages, i.e., they are able to deliver P packets in time O(P+d) where d is the delay on the link.In this paper we examine the effect of eliminating one or both of these assumptions. In particular, we provide an efficient simulation of a linear array of homogeneous processors connected by unit-delay links on a linear array of heterogeneous processors connected by links with arbitrary delay. We show that the slowdown achieved by our simulation is optimal. We then consider the case of simulating cliques by cliques; i.e., a clique of heterogeneous processors with arbitrary delay links is used to simulate a clique of homogeneous processors with unit delay links. We reduce the slowdown from the obvious bound of the maximum delay link to the average of the link delays. In the case of the linear array we consider both links with and without pipelining. For the clique simulation the links are not assumed to support pipelining.The main motivation of our results (as was the case with Andrews et al.) is to mitigate the degradation of performance when executing parallel programs designed for different architectures on a network of workstations (NOW). In such a setting it is unlikely that the links provided by the NOW will support pipelining and it is quite probable the processors will be heterogeneous. Combining our result on clique simulation with well-known techniques for simulating shared memory PRAMs on distributed memory machines provides an effective automatic compilation of a PRAM algorithm on a NOW.