Inequivalent Embeddings Research Articles

Planar Stick Indices of Some Knotted Graphs

Two isomorphic graphs can have inequivalent spatial embeddings in 3-space. In this way, an isomorphism class of graphs contains many spatial graph types. A common way to measure the complexity of a spatial graph type is to count the minimum number of straight sticks needed for its construction in 3-space. In this paper, we give estimates of this quantity by enumerating stick diagrams in a plane. In particular, we compute the planar stick indices of knotted graphs with low crossing numbers. We also show that if a bouquet graph or a theta-curve has the property that its proper subgraphs are all trivial, then the planar stick index must be at least seven.

N = 3 conformal supergravity in four dimensions

In this paper we derive the action for N = 3 conformal supergravity in four space-time dimensions. We construct a density formula for N = 3 conformal supergravity based on the superform action principle. Finally, we embed the N = 3 Weyl multiplet in the density formula to obtain the invariant action for N = 3 conformal supergravity. There are two inequivalent embeddings by changing a particular coefficient from real to imaginary. They lead to invariant actions, which will either be the supersymmetrization of the Weyl square term or the Pontryagin density in the eventuality of gauge fixing to Poincaré supergravity. As a consistency check of our formalism, we will show that the supersymmetrization of the Pontryagin density is a total derivative. We will demonstrate this for purely bosonic terms. We will also present the complete action for the supersymmetrization of Weyl square term. We also discuss consistent truncation of N = 4 Weyl multiplet to N = 3 Weyl multiplet and use it for a robust check of our results using the earlier known results in N = 4 conformal supergravity.

On planar embeddings of pseudoplanar affine curves of genus ≤1

We show that any two planar cubic embeddings of a pseudoplanar curve of genus 1 (over complex numbers) are linearly equivalent. We describe the ‘abstract moduli’ of nonsingular affine plane cubic curves with 3 places at infinity. We note some examples of inequivalent planar embeddings of a nonsingular affine curve of genus 0 having 2 places at infinity.

Classification of Links of Small Complexity in the Thickened Torus

We present a complete table of links in the thickened torus T2 × I of complexity at most 4. The table is constructed by the following method. First, we enumerate all abstract quadrivalent graphs with at most four vertices. Then we consider all inequivalent embeddings of these graphs in the torus. After that each vertex of each of the obtained graphs is replaced by a crossing of one of two possible types specifying the over-strand and the under-strand. The words “over” and “under” are understood in the sense of the coordinate of the corresponding point in the interval I. As a result, we obtain a family of link diagrams in the torus. We propose a number of artificial tricks that essentially reduce the enumeration and help to prove rigorously the completeness of the table. We use a generalized version of the Kauffman polynomial to prove that the obtained links are pairwise inequivalent.

Re-embedding structures of 4-connected projective-planar graphs

We identify the structures of 4-connected projective-planar graphs which generate their inequivalent embeddings on the projective plane, showing two series of graphs the number of whose inequivalent embeddings is held by O(n) with respect to the number of its vertices n. © 2010 Wiley Periodicals, Inc. J Graph Theory 68: 213-228, 2011 © 2011 Wiley Periodicals, Inc.

Representation theory for strange attractors

Embeddings are diffeomorphisms between some unseen physical attractor and a reconstructed image. Different embeddings may or may not be equivalent under isotopy. We regard embeddings as representations of the attractor, review the labels required to distinguish inequivalent representations for an important class of dynamical systems, and discuss the systematic ways inequivalent embeddings become equivalent as the embedding dimension increases until there is finally only one "universal" embedding in a suitable dimension.

Panel structures of triangulations on the torus

Applying the notions called the panel structure and the paneled triangulation introduced by Negami et al. [Re-embedding structures of triangulations on closed surfaces, Sci. Rep. Yokohama Nat. Univ. Sect. I, Math. Phys. Chem. 44 (1997) 41–55], we determine all the numbers of inequivalent embeddings of triangulations on the torus, namely 1–6, 8, 10, 12, 14, 16, 24, 48 and 120.

Embeddings of Danielewski surfaces

A Danielewski surface is defined by a polynomial of the form P=xnz−p(y). Define also the polynomial P′=xnz−r(x)p(y) where r(x) is a non-constant polynomial of degree ≤n−1 and r(0)=1. We show that, when n≥2 and deg p(y)≥2, the general fibers of P and P′ are not isomorphic as algebraic surfaces, but that the zero fibers are isomorphic. Consequently, for every non-special Danielewski surface S, there exist non-equivalent algebraic embeddings of S in ℂ3. Using different methods, we also give non-equivalent embeddings of the surfaces xz=(ydn>−1) for an infinite sequence of integers dn. We then consider a certain algebraic action of the orthogonal group \(\mathcal O(2)\) on ℂ4 which was first considered by Schwarz and then studied by Masuda and Petrie, who proved that this action could not be linearized. This was done by comparing the strata of this action to those of the induced tangent space action. Inequivalent embeddings of a certain singular Danielewski surface S in ℂ3 are found. We generalize their result and show how this leads to an example of two smooth algebraic hypersurfaces in ℂ3 which are algebraically non-isomorphic but holomorphically isomorphic.

Re-embedding structures of 5-connected projective-planar graphs

We shall identify the structures of 5-connected projective-planar graphs which generate their inequivalent embeddings on the projective plane and show a short proof of the result, due to Kitakubo [3], that every 5-connected projective-planar nonplanar graph has exactly 1, 2, 3, 4, 6, 9 or 12 inequivalent embeddings on the projective plane.

Non-extendable isomorphisms between affine varieties

In this paper, we report several large classes of affine varieties (over an arbitrary field K of characteristic 0) with the following property: each variety in these classes has an isomorphic copy such that the corresponding isomorphism cannot be extended to an automorphism of the ambient affine space K n . This implies, in particular, that each of these varieties has at least two inequivalent embeddings in K n . The following application of our results seems interesting: we show that lines in K 2 are distinguished among irreducible algebraic retracts by the property of having a unique embedding in K 2.

Singularities of the projections of n-dimensional knots

Let n be aninteger>4. There is a smoothly knotted n-dimensional sphere in (n+2)-space such that the singular point set of its projection in (n+1)-space consists of double points and that the components of the singular point set are two. (The sphere is knotted in the sense that it does not bound any embedded (n+1)-ball in (n+2)-space.) Furthermore, the projection is not the projection of any unknotted sphere in the (n+2)-space. There are two inequivalent embeddings of an n-manifold in the (n+2)-space such that the projection of one of these in (n+1)-space has no double points and the projection of the other has a connected embedded double point set.

Maps in Locally Orientable Surfaces, the Double Coset Algebra, and Zonal Polynomials

AbstractThe genus series is the generating series for the number of maps (inequivalent two-cell embeddings of graphs), in locally orientable surfaces, closed and without boundary, with respect to vertex- and face-degrees, number of edges and genus. A hypermap is a face two-colourable map. An expression for the genus series for (rooted) hypermaps is derived in terms of zonal polynomials by using a double coset algebra in conjunction with an encoding of a map as a triple of matchings. The expression is analogous to the one obtained for orientable surfaces in terms of Schur functions.

The Number of Embeddings of Quadratic [formula omitted]-Lattices

The number of inequivalent primitive embeddings of a quadratic latticeMinto an indefinite even unimodular Z-latticeL, modulo the action of the orthogonal groupsO(L),SO(L), andO′(L), are determined.

Topologically inequivalent embeddings.

Topologically inequivalent embeddings.

Quantization and representation theory of finiteW algebras

In this paper we study the finitely generated algebras underlying $W$ algebras. These so called 'finite $W$ algebras' are constructed as Poisson reductions of Kirillov Poisson structures on simple Lie algebras. The inequivalent reductions are labeled by the inequivalent embeddings of $sl_2$ into the simple Lie algebra in question. For arbitrary embeddings a coordinate free formula for the reduced Poisson structure is derived. We also prove that any finite $W$ algebra can be embedded into the Kirillov Poisson algebra of a (semi)simple Lie algebra (generalized Miura map). Furthermore it is shown that generalized finite Toda systems are reductions of a system describing a free particle moving on a group manifold and that they have finite $W$ symmetry. In the second part we BRST quantize the finite $W$ algebras. The BRST cohomology is calculated using a spectral sequence (which is different from the one used by Feigin and Frenkel). This allows us to quantize all finite $W$ algebras in one stroke. Explicit results for $sl_3$ and $sl_4$ are given. In the last part of the paper we study the representation theory of finite $W$ algebras. It is shown, using a quantum version of the generalized Miura transformation, that the representations of finite $W$ algebras can be constructed from the representations of a certain Lie subalgebra of the original simple Lie algebra. As a byproduct of this we are able to construct the Fock realizations of arbitrary finite $W$ algebras.

Scrawny Cantor sets are not definable by tori

We define a Cantor set C in R3 to be scrawny if for each p E C and each e > 0 there is a 3 > 0 such that for each map f: S1 -Int B(p , a)C there is a map F: D2 -IntB(p, e) such that FIaD2 = f and F-1(C) is finite. We show the existence and explore some of the properties of wild scrawny Cantor sets in R3. We prove, among other things, that wild scrawny Cantor sets in R3 are not definable by solid tori. 0. INTRODUCTION: HISTORICAL BACKGROUND This paper concerns the existence and properties of a class of wild Cantor sets in R3. Any perfect, uncountable, zero-dimensional compact metric space is called a Cantor set, and any two Cantor sets are topologically equivalent. Since the publication of Antoine's Necklace in 1921 [A], it has been known that there are inequivalent embeddings of Cantor sets in R3; that is, there are Cantor sets C1 and C2 in R3 such that, for any homeomorphism h: R3 -R3 h(C1) e h(C2). Definition. Let C be a Cantor set in R3. If there is a homeomorphism h: R3 R3 such that h(C) lies on a straight line then C is called tame. Otherwise, C is wild. Definition. Let C be a Cantor set in R3. Let {14'jn E N} be a sequence of finite collections On = {Mn ,kI < k < m(n)} of disjoint connected PL 3-manifolds-with-boundary Mn k C R3 such that (1) for each positive integer UAn+I C Int Uad; (2) A{Wu ln E N} = C. Then {J } is called a defining sequence for C. The following result is well known. Received by the editors May 30, 1990. 1991 Mathematics Subject Classification. Primary 57M30, 57M35. The contents of this paper were presented at the Fifth International Topology Conference in Dubrovnik, Yugoslavia, in June, 1990, sponsored by the Union of Mathematicians, Physicists, and Astronomers of Yugoslavia. A travel grant from the National Science Foundation and the Association for Women in Mathematics enabled the author to attend the conference. (?) 1992 American Mathematical Society 0002-9939/92 $1.00 + $.25 per page

Inequivalent genus 2 handlebodies in S3 with homeomorphic complement

An infinite family of inequivalent genus 2 handlebodies embedded in S 3 is described, all of whose complements are of the same homeomorphism class. From an alternate point of view, a compact 3-manifold with incompressible boundary of genus 2 is described which has an infinite number of inequivalent embeddings in S 3 with complement a handlebody.

More on the classification of three generation superstring models

Compactification of the ten dimensional heterotic E 8×E 8 string theory on the Calabi-Yau manifold due to Tian and Yau yields three chiral families of quarks and leptons. The interesting four dimensional gauge groups, after symmetry breaking of E 6 through Wilson loops, turn out to be SU(3) c×SU(3) L×SU(3) R and SU(6)×U(1). The classification of all possible models according to their discrete symmetries for the case of standard embedding of (SU(3)) 3 in E 6 had recently been presented. The present paper completes this analysis by presenting the classification of models for the second inequivalent embedding of (SU(3)) 3 in E 6 and for the two inequivalent embeddings of SU(6)×U(1) in E 6. Only the flipped version of SU(6) is realized for the latter models.

Classification of pseudoparticle solutions in gauge theories

We discuss the classification of pseudoparticle solutions obtained using all inequivalent embeddings of SU(2) in a semisimple algebra $G$. We also discuss some simple consequences of this classification on the interaction of the pseudoparticles of Belavin et al.

Inequivalent embeddings of SU(2) and instanton interactions

The fact that SU(2) allows inequivalent embeddings inside larger groups is shown to have some consequences for the interaction of instantons.