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.

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.

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.

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).

TORUS KNOTS EMBODYING CURVATURE AND TORSION IN AN OTHERWISE FEATURELESS CONTINUUM

2011 ◽  
Vol 20 (12) ◽  
pp. 1723-1739 ◽  
Author(s):  
J. S. AVRIN
Keyword(s):  
Elementary Particle ◽  
General Relativity ◽  
Geometric Model ◽  
The Other ◽  
Particle Model ◽  
Torus Knots ◽  
Torus Knot ◽  
The Subject ◽  
Curvature And Torsion ◽  
The Relationship

The subject is a localized disturbance in the form of a torus knot of an otherwise featureless continuum. The knot's topologically quantized, self-sustaining nature emerges in an elementary, straightforward way on the basis of a simple geometric model, one that constrains the differential geometric basis it otherwise shares with General Relativity (GR). Two approaches are employed to generate the knot's solitonic nature, one emphasizing basic differential geometry and the other based on a Lagrangian. The relationship to GR is also examined, especially in terms of the formulation of an energy density for the Lagrangian. The emergent knot formalism is used to derive estimates of some measurable quantities for a certain elementary particle model documented in previous publications. Also emerging is the compatibility of the torus knot formalism and, by extension, that of the cited particle model, with general relativity as well as with the Dirac theoretic notion of antiparticles.

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 LENS SPACE SURGERIES ALONG CERTAIN 2-COMPONENT LINKS RELATED WITH PARK’S RATIONAL BLOW DOWN, AND REIDEMEISTER-TURAEV TORSION

2013 ◽  
Vol 96 (1) ◽  
pp. 78-126 ◽  
Author(s):  
TERUHISA KADOKAMI ◽  
YUICHI YAMADA
Keyword(s):  
Main Tool ◽  
Lens Space ◽  
The Other ◽  
Dehn Surgery ◽  
Rational Homology ◽  
Blow Down ◽  
Alexander Polynomials ◽  
Other Hand ◽  
Turaev Torsion

AbstractWe study lens space surgeries along two different families of 2-component links, denoted by${A}_{m, n} $and${B}_{p, q} $, related with the rational homology$4$-ball used in J. Park’s (generalized) rational blow down. We determine which coefficient$r$of the knotted component of the link yields a lens space by Dehn surgery. The link${A}_{m, n} $yields a lens space only by the known surgery with$r= mn$and unexpectedly with$r= 7$for$(m, n)= (2, 3)$. On the other hand,${B}_{p, q} $yields a lens space by infinitely many$r$. Our main tool for the proof are the Reidemeister-Turaev torsions, that is, Reidemeister torsions with combinatorial Euler structures. Our results can be extended to the links whose Alexander polynomials are same as those of${A}_{m, n} $and${B}_{p, q} $.

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 THE ALEXANDER POLYNOMIAL OF A TORUS KNOT WITH TWISTS

2006 ◽  
Vol 15 (08) ◽  
pp. 1037-1047 ◽  
Author(s):  
HUGH R. MORTON
Keyword(s):  
Infinite Sequence ◽  
Lens Space ◽  
Alexander Polynomial ◽  
Explicit Calculation ◽  
Dehn Surgery ◽  
Alexander Polynomials ◽  
Torus Knot

This note gives an explicit calculation of the doubly infinite sequence Δ(p, q, 2m), m ∈ Z of Alexander polynomials of the (p, q) torus knot with m extra full twists on two adjacent strings, where p and q are both positive. The knots can be presented as the closure of the p-string braids [Formula: see text], where δp = σp-1σp-2 · σ2σ1, or equally of the q-string braids [Formula: see text]. As an application we give conditions on (p, q) which ensure that all the polynomials Δ(p, q, 2m) with |m| ≥ 2 have at least one coefficient a with |a| > 1. A theorem of Ozsvath and Szabo then ensures that no lens space can arise by Dehn surgery on any of these knots. The calculations depend on finding a formula for the multivariable Alexander polynomial of the 3-component link consisting of the torus knot with twists and the two core curves of the complementary solid tori.

scholarly journals On continuous families of geometric Seifert conemanifold structures

2016 ◽  
Vol 25 (14) ◽  
pp. 1650083
Author(s):  
María Teresa Lozano ◽  
José María Montesinos-Amilibia
Keyword(s):  
Knot Theory ◽  
Orbit Space ◽  
Dehn Surgery ◽  
Singular Set ◽  
Torus Knots ◽  
Left Handed ◽  
Torus Knot ◽  
Seifert Manifolds ◽  
Singular Fibers ◽  
Trefoil Knot

In this paper, dedicated to Prof. Lou Kauffman, we determine the Thurston’s geometry possesed by any Seifert fibered conemanifold structure in a Seifert manifold with orbit space [Formula: see text] and no more than three exceptional fibers, whose singular set, composed by fibers, has at most three components which can include exceptional or general fibers (the total number of exceptional and singular fibers is less than or equal to three). We also give the method to obtain the holonomy of that structure. We apply these results to three families of Seifert manifolds, namely, spherical, Nil manifolds and manifolds obtained by Dehn surgery on a torus knot [Formula: see text]. As a consequence we generalize to all torus knots the results obtained in [Geometric conemanifolds structures on [Formula: see text], the result of [Formula: see text] surgery in the left-handed trefoil knot [Formula: see text], J. Knot Theory Ramifications 24(12) (2015), Article ID: 1550057, 38pp., doi: 10.1142/S0218216515500571] for the case of the left handle trefoil knot. We associate a plot to each torus knot for the different geometries, in the spirit of Thurston.

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.

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