BERGE'S KNOTS IN THE FIBER SURFACES OF GENUS ONE, LENS SPACE AND FRAMED LINKS

In 1990, John Berge described several families of knots in the three-dimensional sphere which have non-trivial Dehn surgeries yielding lens spaces. We study a subfamily of them from the view point of resolution of singularity of complex curves and surfaces, Kirby calculus in topology of four-dimensional manifolds and A'Campo's divide knot theory.

Download Full-text

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

Izvestiya VUZ Applied Nonlinear Dynamics ◽

10.18500/0869-6632-2021-29-6-863-868 ◽

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.

Download Full-text

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

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216517500699 ◽

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.

Download Full-text

FOUR-DIMENSIONAL MANIFOLDS CONSTRUCTED BY LENS SPACE SURGERIES ALONG TORUS KNOTS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216512501118 ◽

2012 ◽

Vol 21 (11) ◽

pp. 1250111 ◽

Cited By ~ 1

Author(s):

MOTOO TANGE ◽

YUICHI YAMADA

Keyword(s):

Lens Space ◽

Complete List ◽

Lens Spaces ◽

Dehn Surgery ◽

Integral Coefficient ◽

Torus Knots ◽

Kirby Calculus ◽

Simply Connected

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 determine the complete list (set) of pairs of integral surgeries along distinct torus knots whose resulting manifolds are orientation preserving/reversing homeomorphic lens spaces, and study the closed 4-manifolds constructed as above. The list consists of five sequences. All framed links and Kirby calculus are indexed by integers. As a bi-product, some sequences of embeddings of lens spaces into the standard 4-manifolds are constructed.

Download Full-text

Adjoint twisted Alexander polynomial of twisted Whitehead links

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216518500268 ◽

2018 ◽

Vol 27 (04) ◽

pp. 1850026

Author(s):

Hoang-An Nguyen ◽

Anh T. Tran

Keyword(s):

Knot Theory ◽

Three Dimensional ◽

Adjoint Action ◽

Alexander Polynomial ◽

Reidemeister Torsion ◽

Alexander Polynomials ◽

Whitehead Link ◽

Twisted Alexander Polynomial ◽

Genus One

The adjoint twisted Alexander polynomial has been computed for twist knots [A. Tran, Twisted Alexander polynomials with the adjoint action for some classes of knots, J. Knot Theory Ramifications 23(10) (2014) 1450051], genus one two-bridge knots [A. Tran, Adjoint twisted Alexander polynomials of genus one two-bridge knots, J. Knot Theory Ramifications 25(10) (2016) 1650065] and the Whitehead link [J. Dubois and Y. Yamaguchi, Twisted Alexander invariant and nonabelian Reidemeister torsion for hyperbolic three dimensional manifolds with cusps, Preprint (2009), arXiv:0906.1500 ]. In this paper, we compute the adjoint twisted Alexander polynomial and nonabelian Reidemeister torsion of twisted Whitehead links.

Download Full-text

Genus one 1-bridge positions for the trivial knot and cabled knots

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004198002916 ◽

1999 ◽

Vol 125 (1) ◽

pp. 53-65 ◽

Cited By ~ 17

Author(s):

CHUICHIRO HAYASHI

Keyword(s):

Solid Torus ◽

Lens Space ◽

Dimensional Sphere ◽

3 Dimensional ◽

Genus One

Let (Vi, ti) be a pair consisting of a solid torus Vi and an unknotted arc ti properly imbedded in Vi for i=1 and 2. Taking the union of these pairs we obtain a pair (M, K) of the 3-dimensional sphere or a lens space M and a knot K. We call such a knot K a genus one 1-bridge knot. In this paper we determine when a genus one 1-bridge knot is the trivial knot or a cabled knot.

Download Full-text

Center of Mass Theorem and the Separation of Variables for Two-Body System on a Three-Dimensional Sphere

Nonlinear Phenomena in Complex Systems ◽

10.33581/1561-4085-2020-23-3-306-311 ◽

2020 ◽

Vol 23 (3) ◽

pp. 306-311

Author(s):

Yu. Kurochkin ◽

Dz. Shoukavy ◽

I. Boyarina

Keyword(s):

Constant Curvature ◽

Jacobi Equation ◽

Separation Of Variables ◽

Center Of Mass ◽

Three Dimensional ◽

Relative Distance ◽

Dimensional Sphere ◽

Body System ◽

Spaces Of Constant Curvature ◽

Hamilton Jacobi Equation

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.

Download Full-text

EXCITON INSULATOR STATES IN MATERIALS WITH â€˜MEXICAN HATâ€™ BAND STRUCTURE DISPERSION AND IN GRAPHENE

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v11i1.6947 ◽

2015 ◽

Vol 11 (1) ◽

pp. 2927-2949

Author(s):

Lyubov E. Lokot

Keyword(s):

Band Structure ◽

Three Dimensional ◽

Dimensional Sphere ◽

Electron Hole ◽

Dinger Equation ◽

Fermi Sphere ◽

Zinc Blende ◽

Bulk Gan ◽

Structure Dispersion ◽

Excitonic States

In the paper a theoretical study the both the quantized energies of excitonic states and their wave functions in grapheneand in materials with "Mexican hat" band structure dispersion as well as in zinc-blende GaN is presented. An integral twodimensionalSchrÃ¶dinger equation of the electron-hole pairing for a particles with electron-hole symmetry of reflection isexactly solved. The solutions of SchrÃ¶dinger equation in momentum space in studied materials by projection the twodimensionalspace of momentum on the three-dimensional sphere are found exactly. We analytically solve an integral twodimensionalSchrÃ¶dinger equation of the electron-hole pairing for particles with electron-hole symmetry of reflection. Instudied materials the electron-hole pairing leads to the exciton insulator states. Quantized spectral series and lightabsorption rates of the excitonic states which distribute in valence cone are found exactly. If the electron and hole areseparated, their energy is higher than if they are paired. The particle-hole symmetry of Dirac equation of layered materialsallows perfect pairing between electron Fermi sphere and hole Fermi sphere in the valence cone and conduction cone andhence driving the Cooper instability. The solutions of Coulomb problem of electron-hole pair does not depend from a widthof band gap of graphene. It means the absolute compliance with the cyclic geometry of diagrams at justification of theequation of motion for a microscopic dipole of graphene where >1 s r . The absorption spectrums for the zinc-blendeGaN/(Al,Ga)N quantum well as well as for the zinc-blende bulk GaN are presented. Comparison with availableexperimental data shows good agreement.

Download Full-text