scholarly journals A POLYNOMIAL INVARIANT FOR LINKS IN LENS SPACES

2012 ◽  
Vol 21 (06) ◽  
pp. 1250060 ◽  
Author(s):  
CHRISTOPHER R. CORNWELL
Keyword(s):  
Lens Space ◽  
Lens Spaces ◽  
Polynomial Invariant ◽  
Grid Diagrams ◽  
Skein Theory ◽  
Skein Relation

We prove the existence of a polynomial invariant that satisfies the HOMFLY skein relation for links in a lens space. In the process we also develop a skein theory of toroidal grid diagrams in a lens space.

scholarly journals The Alexander polynomial of links in lens spaces

2019 ◽  
Vol 28 (08) ◽  
pp. 1950049 ◽  
Author(s):  
E. Horvat ◽  
Boštjan Gabrovšek
Keyword(s):  
Lens Space ◽  
Alexander Polynomial ◽  
Lens Spaces ◽  
Cutting Out ◽  
Skein Relation

We show how the Alexander polynomial of links in lens spaces is related to the classical Alexander polynomial of a link in the 3-sphere, obtained by cutting out the exceptional lens space fiber. It follows from this relationship that a certain normalization of the Alexander polynomial satisfies a skein relation in lens spaces.

Topology of ambient manifolds of non-singular Morse – Smale flows with three periodic orbits

2021 ◽  
Vol 29 (6) ◽  
pp. 863-868
Author(s):  
Danila Shubin ◽  
◽  
Keyword(s):  
Periodic Orbits ◽  
Three Dimensional ◽  
Solid Torus ◽  
Lens Space ◽  
Tubular Neighborhood ◽  
Lens Spaces ◽  
Ambient Manifold ◽  
Orientable Manifold ◽  
Smale Flows ◽  
Published Result

The purpose of this study is to establish the topological properties of three-dimensional manifolds which admit Morse – Smale flows without fixed points (non-singular or NMS-flows) and give examples of such manifolds that are not lens spaces. Despite the fact that it is known that any such manifold is a union of circular handles, their topology can be investigated additionally and refined in the case of a small number of orbits. For example, in the case of a flow with two non-twisted (having a tubular neighborhood homeomorphic to a solid torus) orbits, the topology of such manifolds is established exactly: any ambient manifold of an NMS-flow with two orbits is a lens space. Previously, it was believed that all prime manifolds admitting NMS-flows with at most three non-twisted orbits have the same topology. Methods. In this paper, we consider suspensions over Morse – Smale diffeomorphisms with three periodic orbits. These suspensions, in turn, are NMS-flows with three periodic trajectories. Universal coverings of the ambient manifolds of these flows and lens spaces are considered. Results. In this paper, we present a countable set of pairwise distinct simple 3-manifolds admitting NMS-flows with exactly three non-twisted orbits. Conclusion. From the results of this paper it follows that there is a countable set of pairwise distinct three-dimensional manifolds other than lens spaces, which refutes the previously published result that any simple orientable manifold admitting an NMS-flow with at most three orbits is lens space.

scholarly journals Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology

2008 ◽  
Vol 2008 ◽  
Author(s):  
Kenneth L. Baker ◽  
J. Elisenda Grigsby ◽  
Matthew Hedden
Keyword(s):  
Floer Homology ◽  
Lens Spaces ◽  
Knot Floer Homology ◽  
Grid Diagrams

scholarly journals On the generalization of Conway algebra

2018 ◽  
Vol 27 (02) ◽  
pp. 1850014 ◽  
Author(s):  
Seongjeong Kim
Keyword(s):  
Algebraic Structure ◽  
The Other ◽  
Second Step ◽  
Polynomial Invariant ◽  
Binary Operations ◽  
Other Hand ◽  
Skein Relation

In [Przytyski and Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1989) 115–139], Przytyski and Traczyk introduced an algebraic structure, called a Conway algebra, and constructed an invariant of oriented links, which is a generalization of the Homflypt polynomial invariant. On the other hand, in [Kauffman and Lambropoulou, New invariants of links and their state sum models, arXiv:1703.03655v2 [math.GT] 15 Mar 2017], Kauffman and Lambropoulou introduced new 4-variable invariants of oriented links, which are obtained by two computational steps: in the first step, we apply a skein relation on every mixed crossing to produce unions of unlinked knots. In the second step, we apply another skein relation on crossings of the unions of unlinked knots, which introduces a new variable. In this paper, we will introduce a generalization of the Conway algebra [Formula: see text] with two binary operations and we construct an invariant valued in [Formula: see text] by applying those two binary operations to mixed crossings and pure crossing, respectively. The 4-variable invariant of Kauffman and Lambropoulou with a specific condition is derived from the invariant valued in [Formula: see text]. Moreover, the generalized Conway algebra gives us an invariant of oriented links, which satisfies nonlinear skein relations.

scholarly journals A deformation of the Alexander polynomials of knots yielding lens spaces

2007 ◽  
Vol 75 (1) ◽  
pp. 75-89 ◽  
Author(s):  
Teruhisa Kadokami ◽  
Yuichi Yamada
Keyword(s):  
Lens Space ◽  
Alexander Polynomial ◽  
Lens Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Reidemeister Torsion ◽  
Alexander Polynomials ◽  
Torus Knot ◽  
Necessary And Sufficient

For a knot K in a homology 3-sphere Σ, by Σ(K;p/q), we denote the resulting 3-manifold of p/q-surgery along K. We say that the manifold or the surgery is of lens type if Σ(K;p/q) has the same Reidemeister torsion as a lens space.We prove that, for Σ(K;p/q) to be of lens type, it is a necessary and sufficient condition that the Alexander polynomial ΔK(t) of K is equal to that of an (i, j)-torus knot T(i, j) modulo (tp – 1).We also deduce two results: If Σ(K;p/q) has the same Reidemeister torsion as L(p, q') then (1) (2) The multiple of ΣK(tk) over k ∈ (i) is ±tm modulo (tp – 1), where (i) is the subgroup in (Z/pZ)×/{±1} generated by i. Conversely, if a subgroup H of (Z/pZ)×/{±l} satisfying that the product of ΣK(tk) (k ∈ H) is ±tm modulo (tp – 1), then H includes i or j.Here, i, j are the parameters of the torus knot above.

scholarly journals Algebraic links in lens spaces

2020 ◽  
pp. 2050066
Author(s):  
Eva Horvat
Keyword(s):  
Orbit Space ◽  
Lens Space ◽  
Complex Variables ◽  
Lens Spaces

The lens space [Formula: see text] is the orbit space of a [Formula: see text]-action on the 3-sphere. We investigate polynomials of two complex variables that are invariant under this action, and thus define links in [Formula: see text]. We study properties of these links, and their relationship with the classical algebraic links. We prove that all algebraic links in lens spaces are fibered, and obtain results about their Seifert genus. We find some examples of algebraic knots in [Formula: see text], whose lift in the [Formula: see text]-sphere is a torus link.

scholarly journals An explicit relation between knot groups in lens spaces and those in S3

2018 ◽  
Vol 27 (08) ◽  
pp. 1850045
Author(s):  
Yuta Nozaki
Keyword(s):  
Fundamental Group ◽  
Lens Space ◽  
Lens Spaces ◽  
Alternative Proof ◽  
Cyclic Covering ◽  
Covering Map ◽  
Explicit Relation

For a cyclic covering map [Formula: see text] between two pairs of a 3-manifold and a knot each, we describe the fundamental group [Formula: see text] in terms of [Formula: see text]. As a consequence, we give an alternative proof for the fact that certain knots in [Formula: see text] cannot be represented as the preimage of any knot in a lens space, which is related to free periods of knots. In our proofs, the subgroup of a group [Formula: see text] generated by the commutators and the [Formula: see text]th power of each element of [Formula: see text] plays a key role.

STEIN FILLINGS OF LENS SPACES

2003 ◽  
Vol 05 (06) ◽  
pp. 967-982 ◽  
Author(s):  
R. HIND
Keyword(s):  
Contact Lens ◽  
Finite Energy ◽  
Lens Space ◽  
Symplectic Manifolds ◽  
Holomorphic Curves ◽  
Lens Spaces ◽  
Stein Manifolds ◽  
Stein Fillings

We describe a foliation by finite energy holomorphic curves of some symplectic manifolds which are constructed from Stein manifolds with Lens space boundaries. One application is that all such Stein manifolds bounded by the same contact Lens space are equivalent up to Stein homotopy.

Four-dimensional manifolds constructed by lens space surgeries of distinct types

2017 ◽  
Vol 26 (11) ◽  
pp. 1750069
Author(s):  
Motoo Tange ◽  
Yuichi Yamada
Keyword(s):  
Fiber Surface ◽  
Lens Space ◽  
The Other ◽  
Lens Spaces ◽  
Dehn Surgery ◽  
Integral Coefficient ◽  
Torus Knots ◽  
Torus Knot ◽  
Simply Connected ◽  
Genus One

A framed knot with an integral coefficient determines a simply-connected 4-manifold by 2-handle attachment. Its boundary is a 3-manifold obtained by Dehn surgery along the framed knot. For a pair of such Dehn surgeries along distinct knots whose results are homeomorphic, it is a natural problem: Determine the closed 4-manifold obtained by pasting the 4-manifolds along their boundaries. We study pairs of lens space surgeries along distinct knots whose lens spaces (i.e. the resulting lens spaces of the surgeries) are orientation-preservingly or -reversingly homeomorphic. In the authors’ previous work, we treated with the case both knots are torus knots. In this paper, we focus on the case where one is a torus knot and the other is a Berge’s knot Type VII or VIII, in a genus one fiber surface. We determine the complete list (set) of such pairs of lens space surgeries and study the closed 4-manifolds constructed as above. The list consists of six sequences. All framed links and handle calculus are indexed by integers.

Sign in / Sign up

Export Citation Format

Share Document