On the Alexander polynomial of lens space knots

2020 ◽  
Vol 275 ◽  
pp. 107124
Author(s):  
Motoo Tange
Keyword(s):  
Lens Space ◽  
Alexander Polynomial

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 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.

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 Multivariable link invariants arising from sl(2|1) and the Alexander polynomial

2007 ◽  
Vol 210 (1) ◽  
pp. 283-298 ◽  
Author(s):  
Nathan Geer ◽  
Bertrand Patureau-Mirand
Keyword(s):  
Alexander Polynomial ◽  
Link Invariants

Knots with the Lens Space Surgery

2011 ◽  
Vol 10 (1) ◽  
pp. 561-570
Author(s):  
Alberto Cavicchioli ◽  
Ilaria Telloni
Keyword(s):  
Lens Space

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.

Lens Space

2004 ◽  
pp. 226-226
Author(s):  
Steven Duplij ◽  
Steven Duplij ◽  
Steven Duplij
Keyword(s):  
Lens Space

Lens Space resonators

2010 ◽  
Author(s):  
Jeffrey M. Pond
Keyword(s):  
Lens Space

The Alexander polynomial of alternating knots

2018 ◽  
pp. 127-144
Author(s):  
Alexander Stoimenow
Keyword(s):  
Alexander Polynomial

scholarly journals $K$- and $KO$-rings of the lens space $L\spn(p\sp2)$ for odd prime $p$

1971 ◽  
Vol 1 (2) ◽  
pp. 273-286 ◽  
Author(s):  
Toshihisa Kawaguchi ◽  
Masahiro Sugawara
Keyword(s):  
Lens Space
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