Search

In this paper, we present two different ways for computing the Kauffman bracket skein module of S 1 × S 2 , KBSM S 1 × S 2 , via braids. We first extend the universal Kauffman bracket type invariant V for knots and links in the Solid Torus ST, which is obtained via a unique Markov trace constructed on the generalized Temperley–Lieb algebra of type B, to an invariant for knots and links in S 1 × S 2 . We do that by imposing on V relations coming from the braid band moves. These moves reflect isotopy in S 1 × S 2 and they are similar to the second Kirby move. We obtain an infinite system of equations, a solution of which is equivalent to computing KBSM S 1 × S 2 . We show that KBSM S 1 × S 2 is not torsion free and that its free part is generated by the unknot (or the empty knot). We then present a diagrammatic method for computing KBSM S 1 × S 2 via braids. Using this diagrammatic method, we also obtain a closed formula for the torsion part of KBSM S 1 × S 2 . [ABSTRACT FROM AUTHOR]

In this paper, we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces L (p , q) , KBSM(L (p , q)), for q ≠ 0. For doing this, we introduce a new concept, that of an unoriented braid. Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the generalized Temperley–Lieb algebra of type B, TL 1 , n , which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket-type invariant, V , for knots and links in ST, via a unique Markov trace constructed on TL 1 , n . The universal invariant V is equivalent to the KBSM(ST). For passing now to the KBSM(L (p , q)), we impose on V relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in L (p , q) but not in ST, and which reflect the surgery description of L (p , q) , obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM(L (p , q)). We first present the solution for the case q = 1 , which corresponds to obtaining a new basis, ℬ p , for KBSM(L (p , 1)) with (⌊ p / 2 ⌋ + 1) elements. We note that the basis ℬ p is different from the one obtained by Hoste and Przytycki. For dealing with the complexity of the infinite system for the case q > 1 , we first show how the new basis ℬ p of KBSM(L (p , 1)) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case q > 1. The advantage of the braid theoretic approach that we propose for computing skein modules of c.c.o. 3-manifolds, is that the use of braids provides more control on the isotopies of knots and links in the manifolds, and much of the diagrammatic complexity is absorbed into the proofs of the algebraic statements. [ABSTRACT FROM AUTHOR]

We obtain an analog of Weyl's law for the Kohn Laplacian on lens spaces. We also show that two 3-dimensional lens spaces with fundamental groups of equal prime order are isospectral with respect to the Kohn Laplacian if and only if they are CR isometric. [ABSTRACT FROM AUTHOR]

We compute the action of the 2 -skein algebra of the torus on the 2 -skein module of the solid torus. As a result, we show that the 2 -skein modules of lens spaces is spanned by (⌊ p 2 ⌋ + 1) (2 ⌊ p 2 ⌋ + 1) elements. [ABSTRACT FROM AUTHOR]

Tied links in S 3 were introduced by Aicardi and Juyumaya as standard links in S 3 equipped with some non-embedded arcs, called ties, joining some components of the link. Tied links in the Solid Torus were then naturally generalized by Flores. In this paper, we study this new class of links in other topological settings. More precisely, we study tied links in the lens spaces L (p , 1) , in handlebodies of genus g , and in the complement of the g -component unlink. We introduce the tied braid monoids T M g , n by combining the algebraic mixed braid groups defined by Lambropoulou and the tied braid monoid, and we formulate and prove analogues of the Alexander and the Markov theorems for tied links in the 3-manifolds mentioned above. We also present an L -move braid equivalence for tied braids and we discuss further research related to tied links in knot complements and c.c.o. 3-manifolds. The theory of tied links has potential use in some aspects of molecular biology. [ABSTRACT FROM AUTHOR]

We consider the question of when a rational homology $3$ -sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology group injects in the first homology group of any other element in the same class. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite-rank cokernels. Further consequences include a divisibility condition between the determinants of a connected sum of $2$ -bridge knots and any other knot in the same concordance class. Lastly, we use knot Floer homology combined with our main result to obstruct Dehn surgeries on knots from being rationally cobordant to lens spaces. [ABSTRACT FROM AUTHOR]

In this article, we give a sufficient and necessary condition to that a degree ± 1 self-map on a (2 n - 1) -dimensional lens space for n ≥ 2 is homotopic to a self-homeomorphism. In fact, we show how the endomorphism induced by a homeomorphism can act on the fundamental group of the lens space. Moreover, we provide a specific description of a class of lens spaces which admit self-homeomorphisms inducing nontrivial automorphisms on the fundamental groups. Furthermore, the topologically conjugate classification for a special class of periodic homeomorphisms on S 2 n - 1 is obtained. [ABSTRACT FROM AUTHOR]

We prove that, in order to derive the HOMFLYPT skein module of the lens spaces L (p , 1) from the HOMFLYPT skein module of the solid torus, 𝒮 (ST) , it suffices to solve an infinite system of equations obtained by imposing on the Lambropoulou invariant X for knots and links in the solid torus, braid band moves that are performed only on the first moving strand of elements in a set Λ aug , augmenting the basis Λ of 𝒮 (ST). [ABSTRACT FROM AUTHOR]

In this paper, we give two formulae of values of the Casson–Walker invariant of 3-manifolds with genus one open book decompositions; one is a formula written in terms of a framed link of a surgery presentation of such a 3-manifold, and the other is a formula written in terms of a representation of the mapping class group of a 1-holed torus. For the former case, we compute the invariant through the combinatorial calculation of the degree 1 part of the LMO invariant. For the latter case, we construct a representation of a central extension of the mapping class group through the action of the degree 1 part of the LMO invariant on the space of Jacobi diagrams on two intervals, and compute the invariant as the trace of the representation of a monodromy of an open book decomposition. [ABSTRACT FROM AUTHOR]

In this study, we first classify all topological torus knots lying on the Heegaard torus in lens spaces, and then we study Legendrian representatives of these knots. We classify oriented positive Legendrian torus knots in the universally tight contact structures on the lens spaces up to contactomorphism. [ABSTRACT FROM AUTHOR]

Dehn surgery on a knot determines a dual knot in the surgered manifold, the core of the filling torus. We consider duals of knots in S 3 that have a lens space surgery. Each dual supports a contact structure. We show that if a universally tight contact structure is supported, then the dual is in the same homology class as the dual of a torus knot. [ABSTRACT FROM AUTHOR]

In this paper, the properties of the Kauffman bracket skein module (KBSM) of are investigated. Links in lens spaces are represented both through band and disk diagrams. The possibility to transform between the diagrams enables us to compute the KBSM on an interesting class of examples consisting of inequivalent links with equivalent lifts in the -sphere. The computations show that the KBSM is an essential invariant, that is, it may take different values on links with equivalent lifts. We also show how the invariant is related to the Kauffman bracket of the lift in the -sphere. [ABSTRACT FROM AUTHOR]

We introduce the concept of regular diagrams for knots and links in lens spaces, proving that two diagrams represent equivalent links if and only if they are related by a finite sequence of seven Reidemester type moves. As a particular case, we obtain diagrams and moves for links in ℝℙ3 previously introduced by Y.V. Drobothukina.

We present a new description of the spectrum of the (spin-) Dirac operator D on lens spaces. Viewing a spin lens space L as a locally symmetric space $$\Gamma \backslash {\text {Spin}}(2m)/{\text {Spin}}(2m-1)$$ and exploiting the representation theory of the $${\text {Spin}}$$ groups, we obtain explicit formulas for the multiplicities of the eigenvalues of D in terms of finitely many integer operations. As a consequence, we present conditions for lens spaces to be Dirac isospectral. Tackling classic questions of spectral geometry, we prove with the tools developed that neither spin structures nor isometry classes of lens spaces are spectrally determined by giving infinitely many examples of finite families of Dirac isospectral lens spaces. These results are complemented by examples found with the help of a computer. [ABSTRACT FROM AUTHOR]

In this paper, we work toward the Homflypt skein module of the lens spaces , using braids. In particular, we establish the connection between , the Homflypt skein module of the solid torus ST, and and arrive at an infinite system, whose solution corresponds to the computation of . We start from the Lambropoulou invariant for knots and links in ST, the universal analog of the Homflypt polynomial in ST, and a new basis, , of presented in [I. Diamantis and S. Lambropoulou, A new basis for the Homflypt skein module of the solid torus, J. Pure Appl. Algebra 220(2) (2016) 577-605, , arXiv:1412.3642 [math.GT]]. We show that is obtained from by considering relations coming from the performance of braid band move(s) [bbm] on elements in the basis , where the bbm are performed on any moving strand of each element in . We do that by proving that the system of equations obtained from diagrams in ST by performing bbm on any moving strand is equivalent to the system obtained if we only consider elements in the basic set . The importance of our approach is that it can shed light on the problem of computing skein modules of arbitrary c.c.o. -manifolds, since any -manifold can be obtained by surgery on along unknotted closed curves. The main difficulty of the problem lies in selecting from the infinitum of band moves some basic ones and solving the infinite system of equations. [ABSTRACT FROM AUTHOR]

We construct a positive allowable Lefschetz fibration over the disk on any minimal (weak) symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the canonical contact structure on a lens space is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding complex two-dimensional cyclic quotient singularity. [ABSTRACT FROM AUTHOR]

We study mappings on sub-Riemannian manifolds which are quasiregular with respect to the Carnot-Carathéodory distances and discuss several related notions. On H-type Carnot groups, quasiregular mappings have been introduced earlier using an analytic definition, but so far, a good working definition in the same spirit is not available in the setting of general sub-Riemannian manifolds. In the present paper we adopt therefore a metric rather than analytic viewpoint. As a first main result, we prove that the sub-Riemannian lens space admits nontrivial uniformly quasiregular (UQR) mappings, that is, quasiregular mappings with a uniform bound on the distortion of all the iterates. In doing so, we also obtain new examples of UQR maps on the standard sub-Riemannian spheres. The proof is based on a method for building conformal traps on sub-Riemannian spheres using quasiconformal flows, and an adaptation of this approach to quotients of spheres. One may then study the quasiregular semigroup generated by a UQR mapping. In the second part of the paper we follow Tukia to prove the existence of a measurable conformal structure which is invariant under such a semigroup. Here, the conformal structure is specified only on the horizontal distribution, and the pullback is defined using the Margulis-Mostow derivative (which generalizes the classical and Pansu derivatives). [ABSTRACT FROM AUTHOR]

In this paper we provide algebraic mixed braid classification of links in any c.c.o. 3-manifold M obtained by rational surgery along a framed link in S 3 . We do this by representing M by a closed framed braid in S 3 and links in M by closed mixed braids in S 3 . We first prove an analogue of the Reidemeister theorem for links in M . We then give geometric formulations of the mixed braid equivalence using the L -moves and the braid band moves. Finally we formulate the algebraic braid equivalence in terms of the mixed braid groups B m , n , using cabling and the parting and combing techniques for mixed braids. Our results set a homogeneous algebraic ground for studying links in 3-manifolds and in families of 3-manifolds using computational tools. We provide concrete formuli of the braid equivalences in lens spaces, in Seifert manifolds, in homology spheres obtained from the trefoil and in manifolds obtained from torus knots. Our setting is appropriate for constructing Jones-type invariants for links in families of 3-manifolds via quotient algebras of the mixed braid groups B m , n , as well as for studying skein modules of 3-manifolds, since they provide a controlled algebraic framework and much of the diagrammatic complexity has been absorbed into the proofs. Further, our moves can be used in a braid analogue of Rolfsen's rational calculus and potentially in computing Witten invariants. [ABSTRACT FROM AUTHOR]

We compute the HOMFLYPT skein module of the lens spaces Lb, 1 and present a free basis of this module for each p. [ABSTRACT FROM AUTHOR]

In this paper, we calculate the values of the E6 state sum invariant for the lens spaces L(p, q). In particular, we show that the values of the invariant are determined by p 12 and q (p, 12). As a corollary, we show that the E6 state sum is a homotopy invariant for the oriented lens spaces. [ABSTRACT FROM AUTHOR]

An important geometric invariant of links in lens spaces is the lift in the 3-sphere of a link L in L(p,q), that is the counterimage ... of L under the universal covering of L(p,q). If lens spaces are defined as a lens with suitable boundary identifications, then a link in L(p,q) can be represented by a disk diagram, that is to say, a regular projection of the link on a disk. Starting from this diagram of L, we obtain a diagram of the lift L in S³. Using this construction, we are able to find different knots and links in L(p,q) having equivalent lifts, that is to say, we cannot distinguish different links in lens spaces only from their lift. [ABSTRACT FROM AUTHOR]

In this article we give combinatorial criteria to decide whether a transitive cyclic combinatorial d-manifold can be generalized to an infinite family of such complexes, together with an explicit construction in the case that such a family exists. In addition, we substantially extend the classification of combinatorial 3-manifolds with transitive cyclic symmetry up to 22 vertices. Finally, a combination of these results is used to describe new infinite families of transitive cyclic combinatorial manifolds and in particular a family of neighborly combinatorial lens spaces of infinitely many distinct topological types. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study some aspects of knots and links in lens spaces. Namely, if we consider lens spaces as quotient of the unit ball with suitable identification of boundary points, then we can project the links on the equatorial disk of , obtaining a regular diagram for them. In this context, we obtain a complete finite set of Reidemeister type moves establishing equivalence, up to ambient isotopy, a Wirtinger type presentation for the fundamental group of the complement of the link and a diagrammatic method giving the first homology group. We also compute Alexander polynomial and twisted Alexander polynomials of this class of links, showing their correlation with Reidemeister torsion. [Copyright &y& Elsevier]

Floer constructed instanton homology for integral homology three-spheres. In this paper, we extend instanton Floer homology to lens spaces L( p, q). Moreover we show a gluing formula for a variant of Donaldson invariant along lens spaces. As an application, we prove that $${X = \mathbb {CP}^2 \# \mathbb {CP}^2}$$ does not admit a decomposition $${X = X_1 \cup X_2}$$ . Here X and X are oriented, simply connected, non-spin four-manifolds with b = 1 and with boundary L( p, 2), and p is a prime number of the form 16 N + 1. [ABSTRACT FROM AUTHOR]

Abstract: Let be a symplectic manifold and be a Lagrangian submanifold diffeomorphic to , , or a Lens space of a certain type. Using the symplectic cut and symplectic sum constructions, we express the open Gromov–Witten invariants of in terms of open Gromov–Witten invariants of a pair determined by and the standard Gromov–Witten invariants of a symplectic manifold determined by . We also describe other applications of this approach. [Copyright &y& Elsevier]

We give the explicit formula of the spin-refined Reshetikhin-Turaev SU(2) invariants of lens spaces. Using this result, we also give the formula of the spin-refined Turaev-Viro SU(2) invariants of lens spaces. [ABSTRACT FROM AUTHOR]

We show that if M is a hyperbolic 3-manifold with ∂ M a torus such that M( r) is a lens space and M( r) is toroidal, then Δ( r, r) ≤ 4. [ABSTRACT FROM AUTHOR]

Let K be a two-bridge knot or link in S3. Then K is also denoted as the four-plat, b(p, q) to indicate its association with some rational number p/q. The lens space L = L(p, q) admits an isometry τ of order two, such that the quotient space L modulo the involution τ is an orbifold whose exceptional set is K and whose underlying space is S3. In this paper, the mapping class groups of these orbifolds are classified. While these groups can be found as a result of Mostow's Rigidity Theorem, this paper calculates the generators and relations of the groups and the proof does not rely on this strong theorem for the majority of cases. [ABSTRACT FROM AUTHOR]

Using elementary number theory, we prove several results about the complexity of CR mappings between spheres. It is known that CR mappings between spheres, invariant under finite groups, lead to sharp bounds for degree estimates on real polynomials constant on a hyperplane. We show here that there are infinitely many degrees for which the uniqueness of sharp examples fails. The proof uses a Pell equation. We then sharpen our results and obtain various congruences guaranteeing nonuniqueness. We also show that a gap phenomenon for proper mappings between balls does not occur beyond a certain target dimension. This proof uses the solution of the postage stamp problem. [ABSTRACT FROM AUTHOR]

In this paper we study the Carnot-Caratheodory metrics on SU(2) ≃ S3, SO(3), and SL(2) induced by their Cartan decomposition and by the Killing form. Besides computing explicitly geodesics and conjugate loci, we compute the cut loci (globally), and we give the expression of the Carnot-Caratheodory distance as the inverse of an elementary function. We then prove that the metric given on SU(2) projects on the so-called lens spaces L(p, q). Also for lens spaces, we compute the cut loci (globally). For SU(2) the cut locus is a maximal circle without one point. In all other cases the cut locus is a stratified set. To our knowledge, this is the first explicit computation of the whole cut locus in sub-Riemannian geometry, except for the trivial case of the Heisenberg group. [ABSTRACT FROM AUTHOR]

The quantum SO(3)-invariants of ℚ-homology 3-spheres can be perturbatively expanded to the Ohtsuki invariant in number theory. On the other hand, it is known that the quantum SU(2)-invariants of 3-manifolds M admits a refinement involving a mod 2 cohomology class of M. A motivation of this paper is to study whether a perturbative expansion can be derived from the refinement. In this paper, we shall prove to be able to derive the perturbative expansion in the case of lens spaces by a concrete calculation, and shall define the perturbative invariant. Furthermore we obtain some properties for the perturbative invariant for covering spaces and connected sums of lens spaces. [ABSTRACT FROM AUTHOR]

We first prove a uniqueness result for certain group-invariant CR mappings to hyperquadrics. For cyclic groups these mappings lead to a collection of polynomials ƒp,q (with integer coefficients) in two variables; here p and q are positive integers. We use the uniqueness result to prove some surprising number-theoretic results about the ƒp,q, in particular, ƒp,q is congruent to xP + yP modulo (p) (for P ≥ 2) if and only if p is prime. We also determine recurrence relations for these polynomials for q ≤ 3 and determine a functional equation for a generating function. Finally, we discuss the invariant polynomials that arise for certain representations of dihedral groups to illustrate the non-Abelian case. [ABSTRACT FROM AUTHOR]

The simplest NMS flow is a polar flow formed by an attractive periodic orbit and a repulsive periodic orbit as limit sets. In this paper we show that the only orientable, simple, compact, 3-dimensional manifolds without boundary that admit an NMS flow with none or one saddle periodic orbit are lens spaces. We also see that when a fattened round handle is a connected sum of tori, the corresponding flow is also a trivial connected sum of flows. [ABSTRACT FROM AUTHOR]

Let K be a two-bridge knot in S[sup 3]. Then K is also denoted as the four-plat, b(p,q) to indicate its association with some rational number p/q. The lens space L = L(p,q) admits an isometry, τ, of order 2, such that the quotient space L modulo the involution τ is an orbifold whose exceptional set is K. In this paper, the isometries of these orbifolds are classified; this is equivalent to computing the isometries of (S[sup 3],K). [ABSTRACT FROM AUTHOR]

Let α(d) denote the number of ones in the binary expansion of d. For 1⩽k⩽α(d) we prove that the 2(d+α(d)−k)+1-dimensional, 2k-torsion lens space does not immerse in a Euclidian space of dimension 4d−2α(d) provided certain technical condition holds. The extra hypothesis is easily eliminated in the case k=1 recovering Davis’ strong non-immersion theorem for real projective spaces. For k>1 this is a deeper problem (solved only in part) that requires a close analysis of the interaction between the Brown–Peterson 2-series and its 2k analogue. The methods are based on a partial generalization of the Brown–Peterson version for the Conner–Floyd conjecture used in this context to detect obstructions for the existence of Euclidian immersions. [Copyright &y& Elsevier]