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     

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.     

Nonfacets for shellable spheres

We generalize the notion of a map on a 2-sphere to maps on then-sphere and then show that there exist combinatorial types of countries that cannot be the only type of country for a shellablen-sphere. This generalizes the well known theorem that there are no maps on the 2-sphere all of whose countries arek-gons for anyk≧6. Research supported by N.S.F. grant, number GP-42941     

A perturbative SU(3) Casson invariant

A perturbative SU(3) Casson invariant for integral homology 3-sphere is defined. Besides being fully perturbative, it has the nice properties: (1) is an integer. (2) It is preserved under orientation change. (3) A connect sum formula. Explicit calculations of the invariant for 1/k surgery of (2, q) torus knot are presented and compared with Boden-Herald‚s different SU(3) generalization of Casson‚s invariant. For those cases computed, the invariant defined here is a quadratic polynomial in k for k > 0 and a different quadratic polynomial for k < 0. Received: October 12, 2000     

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.     

Transitive Factorizations in the Symmetric Group, and Combinatorial Aspects of Singularity Theory

We consider the determination of the number c_k(α) of ordered factorizations of an arbitrary permutation on n symbols, with cycle distribution α, intok -cycles such that the factorizations have minimal length and the group generated by the factors acts transitively on then symbols. The case k =  2 corresponds to the celebrated result of Hurwitz on the number of topologically distinct holomorphic functions on the 2-sphere that preserve a given number of elementary branch point singularities. In this case the monodromy group is the full symmetric group. For k =  3, the monodromy group is the alternating group, and this is another case that, in principle, is of considerable interest. We conjecture an explicit form, for arbitrary k, for the generating series for c_k(α), and prove that it holds for factorizations of permutations with one, two and three cycles (so α is a partition with at most three parts). Our approach is to determine a differential equation for the generating series from a combinatorial analysis of the creation and annihilation of cycles in products under the minimality condition.     

A note on graphs and sphere orders

A partially ordered set P is called a k-sphere order if one can assign to each element a ∈ P a ball B_a in R^k so that a < b iff B_a ? B_b. To a graph G = (V,E) associate a poset P(G) whose elements are the vertices and edges of G. We have v < e in P(G) exactly when v ∈ V, e ∈ E, and v is an end point of e. We show that P(G) is a 3-sphere order for any graph G. It follows from E. R. Scheinerman [“A Note on Planar Graphs and Circle Orders,” SIAM Journal of Discrete Mathematics, Vol. 4 (1991), pp. 448–451] that the least k for which G embeds in R^k equals the least k for which P(G) is a k-sphere order. For a simplicial complex K one can define P(K) by analogy to P(G) (namely, the face containment order). We prove that for each 2-dimensional simplicial complex K, there exists a k so that P(K) is a k-sphere order. © 1993 John Wiley & Sons, Inc.     

A geometric formula for Haefliger knots

Haefliger has shown that a smooth embedding of the (4k−1)-sphere in the 6k-sphere can be knotted in the smooth sense. In this paper, we give a formula with which we can detect the isotopy class of such a Haefliger knot. The formula is expressed in terms of the geometric characteristics of an extension, analogous to a Seifert surface, of the given embedding. In particular, the Hopf invariant associated to the extension plays a crucial role. This leads us to a new characterisation of Haefliger knots.     

Geometric formulas for Smale invariants of codimension two immersions

We give three formulas expressing the Smale invariant of an immersion f of a (4k−1)-sphere into (4k+1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where g restricted to the boundary is an immersion regularly homotopic to f in (6k−1)-space.The formulas imply that if f and g are two non-regularly homotopic immersions of a (4k−1)-sphere into (4k+1)-space then they are also non-regularly homotopic as immersions into (6k−1)-space. Moreover, any generic homotopy in (6k−1)-space connecting f to g must have at least a_k(2k−1)! cusps, where a_k=2 if k is odd and a_k=1 if k is even.     

Complementary Regions of Knot and Link Diagrams

An increasing sequence of integers is said to be universal for knots and links if every knot and link has a reduced projection on the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. In this paper, it is proved that the following infinite sequences are each universal for knots and links: (3, 5, 7, . . .), (2, n, n + 1, n + 2, . . .) for each n ≥ 3, (3, n, n + 1, n + 2, . . .) for each n ≥ 4. Moreover, the finite sequences (2, 4, 5) and (3, 4, n) for each n ≥ 5 are universal for all knots and links. It is also shown that every knot has a projection with exactly two odd-sided faces, which can be taken to be triangles, and every link of n components has a projection with at most n odd-sided faces if n is even and n + 1 odd-sided faces if n is odd.     

Sphere and Dot Product Representations of Graphs

A graph G is a k-sphere graph if there are k-dimensional real vectors v ₁,…,v _n such that ij∈E(G) if and only if the distance between v _i and v _j is at most 1. A graph G is a k-dot product graph if there are k-dimensional real vectors v ₁,…,v _n such that ij∈E(G) if and only if the dot product of v _i and v _j is at least 1.     

Eigenvalues and suspension structure of compact Riemannian orbifolds with positive Ricci curvature

Let M be a compact n-dimensional Riemannian orbifold of Ricci curvature ≥n−1. We prove that for 1 ≤k≤n, the k ^th nonzero eigenvalue of the Laplacian on M is equal to the dimension n if and only if M is isometric to the k-times spherical suspension over the quotient S ^{n − k} }Γ of the unit (n−k)-sphere by a finite group Γ⊂O(n−k+1) acting isometrically on S ^{n − k} ⊂ℝ ^{n − k +}. Received: 21 September 1998 / Revised version: 23 February 1999     

Modal Aggregation and the Theory of Paraconsistent Filters

This paper articulates the structure of a two species of weakly aggregative necessity in a common idiom, neighbourhood semantics, using the notion of a k-filter of propositions. A k-filter on a non-empty set I is a collection of subsets of I which (i) contains I, (ii) is closed under supersets on I, and (iii) contains ∪{X_i ≤ X_j : 0 ≤ i < j ≤ k} whenever it contains the subsets X₀,…, X_k. The mathematical content of the proof that weakly aggregative modal logic is complete relative to k-ary frame theory, the standard semantic idiom for weakly aggregative modal logic (see [1]) is presented in language-independent terms as a representation theorem for k-filters: every non-trivial k-filter is included in the union of ≤ k non-trivial filters. The elementary theory of k-filters is developed and then applied in the form of an ultrafilter extension result for k-ary frame theory. Mathematics Subject Classification: 03B45.     

Link concordance invariants and homotopy theory

Summary We show that each of a sequence of concordance invariants for codimension-two links of spheres inS ⁿ⁺² , defined by Kent Orr, is identically zero forn>1. For classical links (n=1), the same proof shows that these invariants vanish if and only if Milnor's vanish (a result obtained independently and earlier by Orr himself). We offer sufficient conditions for the vanishing of Orr's -invariant (not covered by the above). We discuss how this relates to positive results.Supported by a National Science Foundation Postdoctoral Fellowship and the hospitality of Univ. Calif. San Diego Math. Dept.     

The links of 3‐dimensional singularities defined by Brieskorn polynomials

In this paper, we completely determine the diffeomorphism types of the 5‐dimensional links of 3‐dimensional log‐canonical singularities defined by Brieskorn polynomials. Moreover, we show that if k is an integer with 1 ≤ k < 611, then there is no link K defined by a Brieskorn polynomial in ?⁴ such that the order of H₂(K) is 6k. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Métriques autoduales sur la boule

A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of the sphere. In this paper we characterize the conformal metrics and trace-free second fundamental forms on the 3-sphere (close to the standard round metric) which are boundaries of selfdual conformal metrics on the whole 4-ball. When the data on the boundary is reduced to a conformal metric (the trace-free part of the second fundamental form vanishes), one may hope to find in the conformal class of the filling metric an Einstein metric, with a pole of order 2 on the boundary. We determine which conformal metrics on the 3-sphere are boundaries of such selfdual Einstein metrics on the 4-ball. In particular, this implies the Positive Frequency Conjecture of LeBrun. The proof uses twistor theory, which enables to translate the problem in terms of complex analysis; this leads us to prove a criterion for certain integrable CR structures of signature (1,1) to be fillable by a complex domain. Finally, we solve an analogous, higher dimensional problem: selfdual Einstein metrics are replaced by quaternionic-K?hler metrics, and conformal structures on the boundary by quaternionic contact structures (previously introduced by the author); in contrast with the 4-dimensional case, we prove that any small deformation of the standard quaternionic contact structure on the (4m−1)-sphere is the boundary of a quaternionic-K?hler metric on the (4m)-ball. Oblatum 29-XI-2000 & 7-XI-2001?Published online: 1 February 2002     

On the Homflypt skein module of $S^1 \times S^2$

Let k be a subring of the field of rational functions in x, v,s which contains . If M is an oriented 3-manifold, let denote the Homflypt skein module of M over k. This is the free k-module generated by isotopy classes of framed oriented links in M quotiented by the Homflypt skein relations: (1) ; (2) L with a positiv e twist ; (3) where O is the unknot. We give two bases for the relative Homflypt skein module of the solid torus with 2 points in the boundary. The first basis is related to the basis of given by J. Hoste and M. Kidwell and also V. Turaev; the second basis is related to a Young idempotent basis for based on the work of A. Aiston, H. Morton and C. Blanchet. We prove that if the elements , for n a nonzero integer, and the elements , for any integer m, are invertible ink, then -torsion module . Here the free part is generated by the empty link . In addition, if the elements , for m an integer, are invertible in k, then has no torsion. We also obtain some results for more general k. Received January 7, 2000; in final form September 20, 2000 / Published online April 12, 2001     

Interval Routing onk-Trees

A graph has an optimall-interval routing scheme if it is possible to direct messages along shortest paths by labeling each edge with at mostlpairwise-disjoint subintervals of the cyclic interval [1…n] (where each node of the graph is labeled by an integer in the range). Although much progress has been made forl = 1, there is as yet no general tight characterization of the classes of graphs associated with largerl. Bodlaenderet al. have shown that under the assumption of dynamic cost links, each graph with an optimall-interval routing scheme has treewidth of at most 4l. For the setting without dynamic cost links, this paper addresses the complementary question of the number of intervals required to label classes of graphs of treewidthk. Although it has been shown that there exist graphs of treewidth 2 that require a nonconstant number of intervals, our work demonstrates a class of graphs of treewidth 2, namely 2-trees, that are guaranteed to allow 3-interval routing schemes. In contrast, this paper presents a 2-tree that cannot have a 2-interval routing scheme. For generalk, anyk-tree is shown to have an optimal interval routing scheme using 2^{k + 1}intervals per edge.     

Tensor Product Representations of the Lie Superalgebra 𝔭(n) and Their Centralizers

Abstract

We construct an associative algebra A _k and show that there is a representation of A _k on V ^?k , where V is the natural 2n-dimensional representation of the Lie superalgebra 𝔭(n). We prove that A _k is the full centralizer of 𝔭(n) on V ^?k , thereby obtaining a “Schur-Weyl duality” for the Lie superalgebra 𝔭(n). This result is used to understand the representation theory of the Lie superalgebra 𝔭(n). In particular, using A _k we decompose the tensor space V ^?k , for k = 2 or 3, and show that V ^?k is not completely reducible for any k ≥ 2.     

Homotopy Spheres in Formal Language

The set of nonempty proper subwords of a word is either contractible or a homotopy n-sphere. There is a simple algorithm which computes n. The existence of spherical words is investigated, and the words which yield spheres are determined. A language which is closed under subwords has a finite number of components, and each component has a finitely generated fundamental group. For each n greater than 1, there is a language on two letters which has the homotopy type of an infinite cluster of n-sphere. There is a language on two letters which has nontrivial homology in each dimension greater than 1. If a language is closed under subwords and has bounded period, then it has the homotopy type of a finite polyhedron.