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

2012 ◽  
Vol 21 (11) ◽  
pp. 1250111 ◽  
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.

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.

Homology of manifolds obtained by Dehn surgery on knots in lens spaces

2000 ◽  
Vol 09 (04) ◽  
pp. 431-442
Author(s):  
Antje Christensen
Keyword(s):  
Homology Group ◽  
Necessary Conditions ◽  
Lens Space ◽  
Lens Spaces ◽  
Dehn Surgery ◽  
Torus Knots

The question whether or not a Dehn surgery on a knot in a lens space yields a lens space of the same order is investigated with homological techniques. Determining the first homology group of the lens space after surgery and of its covering yields some necessary conditions on the knot and the surgery curve. Application of these results along with a calculation of Seifert invariants answers the question completely for surgery on torus knots along nullhomological curves.

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

2005 ◽  
Vol 14 (02) ◽  
pp. 177-188 ◽  
Author(s):  
YUICHI YAMADA
Keyword(s):  
Knot Theory ◽  
Three Dimensional ◽  
Lens Space ◽  
Lens Spaces ◽  
Dimensional Sphere ◽  
View Point ◽  
Complex Curves ◽  
Resolution Of Singularity ◽  
Kirby Calculus ◽  
Genus One

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.

scholarly journals Cyclic surgery on satellite knots

1991 ◽  
Vol 33 (2) ◽  
pp. 125-128 ◽  
Author(s):  
Xingru Zhang
Keyword(s):  
Fundamental Group ◽  
Lens Space ◽  
Dehn Surgery ◽  
Torus Knots ◽  
Torus Knot ◽  
Cyclic Surgery ◽  
Satellite Knots

In [9] L. Moser classified all manifolds obtained by Dehn surgery on torus knots. In particular she proved the following (see also [8, Chapter IV]).Theorem 1 [9]. Nontrivial surgery with slope m/n on a nontrivial torus knot T(p, q) gives a manifold with cyclic fundamental group iff m = npq ± 1 and the manifold obtained is the lens space L(m, nq2).

SPINAL KNOTS IN LENS SPACES

2006 ◽  
Vol 15 (10) ◽  
pp. 1371-1389
Author(s):  
ARNAUD DERUELLE ◽  
DANIEL MATIGNON
Keyword(s):  
Main Part ◽  
Lens Space ◽  
Lens Spaces ◽  
Dehn Surgery ◽  
Intersection Graphs ◽  
Möbius Band

A knot in a lens is said to be spinal if it can be isotoped on a standard spine (e.g. in ℝP3, spinal knots bound a Möbius band). We prove that a Dehn surgery on a non-spinal knot in a lens space cannot produce 𝕊3. With a view to study the Dehn surgeries that produce lens spaces, the main part is devoted to finding an obstruction for a standard spine to be minimal. We consider the intersection graphs coming from a standard spine and an arbitrary surface. This obstruction is given by the existence of a generalized Scharlemann cycle.

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 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 KNOTS WITH DISTINCT PRIMITIVE/PRIMITIVE AND PRIMITIVE/SEIFERT REPRESENTATIVES

2012 ◽  
Vol 21 (01) ◽  
pp. 1250015 ◽  
Author(s):  
BRANDY J. GUNTEL
Keyword(s):  
Lens Space ◽  
Parameter Family ◽  
Surface Slope ◽  
Torus Knots ◽  
Genus 2 ◽  
Similar Class ◽  
Heegaard Surface ◽  
Two Parameter

Berge introduced knots that are primitive/primitive with respect to the genus 2 Heegaard surface, F, in S3; surgery on such knots at the surface slope yields a lens space. Later Dean described a similar class of knots that are primitive/Seifert with respect to F; surgery on these knots at the surface slope yields a Seifert fibered space. Here we construct a two-parameter family of knots that have distinct primitive/Seifert embeddings in F with the same surface slope, as well as a family of torus knots that have a primitive/primitive representative and a primitive/Seifert representative with the same surface slope.

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