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At least 73 records · Page 4Linked to original sources
arXiv2005

Electrodynamics and the Gauss Linking Integral on the 3-Sphere and in Hyperbolic 3-Space

In this first of two papers, we develop a steady-state version of classical electrodynamics on the 3-sphere and in hyperbolic 3-space, including an explicit formula for the vector-valued Green's operator, an explicit formula of Biot-Savart type for the magnetic field, and a corresponding Ampere's Law contained in Maxwell's equations. We then use this to obtain explicit integral formulas for the linking number of two disjoint closed curves in these spaces. All the formulas, like their prototypes in Euclidean 3-space, are geometric rather than just topological, in the sense that their integrands are invariant under orientation-preserving isometries of the ambient space. In the second paper, we obtain integral formulas for twisting, writhing and helicity, and prove the theorem LINK = TWIST + WRITHE in the 3-sphere and in hyperbolic 3-space. We then use these results to derive upper bounds for the helicity of vector fields and lower bounds for the first eigenvalue of the curl operator on subdomains of these two spaces. An announcement of most of these results, and a hint of their proofs, can be found in the Math ArXiv, math.GT/0406276.

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arXiv2007

Multivariable link invariants arising from Lie superalgebras of type I

This paper generalize [7](math.GT/0601291): We construct new links invariants from g, a type I basic classical Lie superalgebra. The construction uses the existence of an unexpected replacement of the vanishing quantum dimension of typical module. Using this, we get a multivariable link invariant associated to any one parameter family of irreducible g-modules.

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arXiv2006

The L-Move and Virtual Braids

This paper is a short introduction to and statement of the main theorems of our paper "Virtual Braids and the L-Move", JKTR, Vol. 15, No. 6 (2006), pp. 773-811. See also arxiv:Math.GT/0507035.

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arXiv2006

Orientations of Chord Diagrams and Khovanov Homology

By adding or removing appropriate structures to Gauss diagram, one can create useful objects related to virtual links. In this paper few objects of this kind are studied: twisted virtual links generalizing virtual links; signed chord diagrams staying halfway between twisted virtual links and Kauffman bracket / Khovanov homology; alternatable virtual links intermediate between virtual and classical links. The most profound role here belongs to a structure that we dare to call orientation of chord diagram. Khovanov homology is generalized to oriented signed chord diagrams and links in oriented thickened surface such that the link projection realizes the first Stiefel-Whitney class of the surface. After this paper was published, V.O.Manturov succeeded in extending Khovanov homology with arbitrary coefficients to arbitrary virtual links, see arXiv: math.GT/0601152.

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arXiv2006

A universal invariant of four-dimensional 2-handlebodies and three-manifolds

This paper is the second part of our work on 4-dimensional 2-handlebodies. In the first part (arXiv:math.GT/0407032) it is shown that up to certain set of local moves, connected simple coverings of B^4 branched over ribbon surfaces, bijectively represent connected orientable 4-dimensional 2-handlebodies up to 2-deformations (handle slides and creations/cancellations of handles of index <= 2). We factor this bijective correspondence through a map onto the closed morphisms in a universal braided category freely generated by a Hopf algebra object H. In this way we obtain a complete algebraic description of 4-dimensional 2-handlebodies. This result is then used to obtain an analogous description of the boundaries of such handlebodies, i.e. 3-dimensional manifolds, which resolves for closed manifolds the problem posed by Kerler in "Towards an algebraic characterization of 3-dimensional cobordisms", Contemporary Mathematics 318 (2003). (cf. Problem 8-16 (1) in T. Ohtsuki, "Problems on invariants of knots and 3-manifolds", Geom. Topol. Monogr. 4 (2002).

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arXiv2008

A note on transversal knots that are closed 3-braids

The contents of this 6-page paper have been subsumed into the 13-page paper, "A note on closed 3-braids", arXiv:0802.1072 [math.GT]. This paper is correct, but contains less information than the new one. The topological classification of knots that are closed braids is shown to lead to a classification theorem for transversal knots that are closed 3-braids. A list is given of all low crossing examples of transversally non-simple knots that are closed 3-braids.

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arXiv2001

Equivariant configuration spaces

We use the compression theorem (arxiv:math.GT/9712235) cf section 7, to prove results for equivariant configuration spaces analogous to the well-known non-equivariant results of May, Milgram and Segal.

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arXiv1999

Computations of Quandle Cocycle Invariants of Knotted Curves and Surfaces

State-sum invariants for knotted curves and surfaces using quandle cohomology were introduced by Laurel Langford and the authors in math.GT/9903135 In this paper we present methods to compute the invariants and sample computations. Computer calculations of cohomological dimensions for some quandles are presented. For classical knots, Burau representations together with Maple programs are used to evaluate the invariants for knot table. For knotted surfaces in 4-space, movie methods and surface braid theory are used. Relations between the invariants and symmetries of knots are discussed.

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arXiv2007

Groups that act pseudofreely on S^2 x S^2

Recall that a pseudofree group action on a space is one whose singular set consists only of isolated points. In this paper, we classify all of the finite groups which admit pseudofree actions on S^2 x S^2. The groups turn out to be exactly the expected ones: those which admit orthogonal pseudofree actions on S^2 x S^2 as a subset of R^3 x R^3. They are explicitly listed. This paper can be viewed as a companion to a paper of Edmonds, math.GT/9809055.

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arXiv2002

Topological Field Theory Interpretation of String Topology

The string bracket introduced by Chas and Sullivan [math.GT/9911159] is reinterpreted from the point of view of topological field theories in the Batalin-Vilkovisky or BRST formalisms. Namely, topological action functionals for gauge fields (generalizing Chern-Simons and BF theories) are considered together with generalized Wilson loops. The latter generate a (Poisson or Gerstenhaber) algebra of functionals with values in the $S^1$-equivariant cohomology of the loop space of the manifold on which the theory is defined. It is proved that, in the case of $GL_n$ with standard representation, the (Poisson or BV) bracket of two generalized Wilson loops applied to two cycles is the same as the generalized Wilson loop applied to the string bracket of the cycles. Generalizations to other groups are briefly described.

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arXiv1999

Structures de contact en dimension trois et bifurcations des feuilletages de surfaces

The main purpose of this article is to classify contact structures on some 3-manifolds, namely lens spaces, most torus bundles over a circle, the solid torus, and the thickened torus T^2 x [0,1]. This classification completes earlier work (by Etnyre [math.DG/9812065], Eliashberg, Kanda, Makar-Limanov, and the author) and results from the combination of two techniques: surgery, which produces many contact structures, and tomography, which allows one to analyse a contact structure given a priori and to create from it a combinatorial image. The surgery methods are based on a theorem of Y. Eliashberg -- revisited by R. Gompf [math.GT/9803019] -- and produces holomorphically fillable contact structures on closed manifolds. Tomography theory, developed in parts 2 and 3, draws on notions introduced by the author and yields a small number of possible models for contact structures on each of the manifolds listed above.

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arXiv2004

Algebraic functions and closed braids

This article was originally published in Topology 22 (1983). The present hyperTeXed redaction includes references to post-1983 results as Addenda, and corrects a few typographical errors. (See math.GT/0411115 for a more comprehensive overview of the subject as it appears 21 years later.)

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arXiv2002

Linking numbers of measured foliations

We generalise the average asymptotic linking number of a pair of divergence-free vector fields on homology three-spheres by considering the linking of a divergence-free vector field on a manifold of arbitrary dimension with a codimension two foliation endowed with an invariant transverse measure. We prove that the average asymptotic linking number is given by an integral of Hopf type. Considering appropriate vector fields and measured foliations, we obtain an ergodic interpretation of the Godbillon-Vey invariant of a family of codimension one foliations discussed in math.GT/0111137.

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arXiv2002

Addendum and correction to: Homology cylinders: an enlargement of the mapping class group

In a previous paper [Homology cylinders: an enlargement of the mapping class group, Algebr. Geom. Topol. 1 (2001) 243--270, arXiv:math.GT/0010247], a group H_g of homology cylinders over the oriented surface of genus g is defined. A filtration of H_g is defined, using the Goussarov-Habiro notion of finite-type. It is erroneously claimed that this filtration essentially coincides with the relative weight filtration. The present note corrects this error and studies the actual relation between the two filtrations.

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