scholarly journals Embedding spheres in knot traces

2021 ◽  
Vol 157 (10) ◽  
pp. 2242-2279
Author(s):  
Peter Feller ◽  
Allison N. Miller ◽  
Matthias Nagel ◽  
Patrick Orson ◽  
Mark Powell ◽  
...  
Keyword(s):  
Fundamental Group ◽  
Homotopy Group ◽  
Alexander Polynomial ◽  
Homotopy Equivalent ◽  
Knot Invariants

Abstract The trace of the $n$ -framed surgery on a knot in $S^{3}$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded $2$ -sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable $3$ -dimensional knot invariants. For each $n$ , this provides conditions that imply a knot is topologically $n$ -shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.

scholarly journals TOPOLOGICAL 4-MANIFOLDS WITH 4-DIMENSIONAL FUNDAMENTAL GROUP

2021 ◽  
pp. 1-8
Author(s):  
DANIEL KASPROWSKI ◽  
MARKUS LAND
Keyword(s):  
Fundamental Group ◽  
Homotopy Equivalent ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Canonical Map ◽  
Duality Space ◽  
Simply Connected Manifolds ◽  
Simply Connected ◽  
Poincaré Duality Space

Abstract Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

Strongly-cyclic branched coverings and the Alexander polynomial of knots in rational homology spheres

2007 ◽  
Vol 142 (2) ◽  
pp. 259-268 ◽  
Author(s):  
YUYA KODA
Keyword(s):  
Fundamental Group ◽  
Homology Group ◽  
Alexander Polynomial ◽  
Homology Sphere ◽  
Cyclic Covering ◽  
Rational Homology ◽  
Branched Coverings ◽  
Branched Covering ◽  
Rational Homology Spheres ◽  
Rational Homology Sphere

AbstractLet K be a knot in a rational homology sphere M. In this paper we correlate the Alexander polynomial of K with a g-word cyclic presentation for the fundamental group of the strongly-cyclic covering of M branched over K. We also give a formula for the order of the first homology group of the strongly-cyclic branched covering.

Sharp Bounds on Some Classical Knot Invariants

2003 ◽  
Vol 12 (06) ◽  
pp. 805-817
Author(s):  
C. Kearton ◽  
S. M. J. Wilson
Keyword(s):  
Alexander Polynomial ◽  
Sharp Bounds ◽  
Knot Invariants ◽  
Bridge Number

There are obvious inequalities relating the Nakanishi index of a knot, the bridge number, the degree 2n of the Alexander polynomial and the length of the chain of Alexander ideals. We give examples for every positive value of n to show that these bounds are sharp.

RECURSION FORMULAS FOR SOME ABELIAN KNOT INVARIANTS

2000 ◽  
Vol 09 (03) ◽  
pp. 413-422 ◽  
Author(s):  
WAYNE H. STEVENS
Keyword(s):  
Recursion Formula ◽  
Alexander Polynomial ◽  
Square Roots ◽  
Knot Invariants ◽  
Homology Groups ◽  
Cyclic Covers ◽  
Recursion Formulas ◽  
Branched Cyclic Covers ◽  
Linear Recursion ◽  
Fold Cover

Let K be a tame knot in S3. We show that the sequence of cyclic resultants of the Alexander polynomial of K satisfies a linear recursion formula with integral coefficients. This means that the orders of the first homology groups of the branched cyclic covers of K can be computed recursively. We further establish the existence of a recursion formula that generates sequences which contain the square roots of the orders for the odd-fold covers that contain the square roots of the orders for the even-fold covers quotiented by the order for the two-fold cover. (That these square roots are all integers follows from a theorem of Plans.)

On the Asphericity of Knot Complements

1993 ◽  
Vol 45 (2) ◽  
pp. 340-356
Author(s):  
Vo Thanh Liem ◽  
Gerard A. Venema
Keyword(s):  
Fundamental Group ◽  
Homotopy Group ◽  
Homotopy Groups ◽  
Unusual Property ◽  
Knot Complements ◽  
Flat Embedding

AbstractTwo examples of topological embeddings of S2 in S4 are constructed. The first has the unusual property that the fundamental group of the complement is isomorphic to the integers while the second homotopy group of the complement is nontrivial. The second example is a non-locally flat embedding whose complement exhibits this property locally.Two theorems are proved. The first answers the question of just when good π1 implies the vanishing of the higher homotopy groups for knot complements in S4. The second theorem characterizes local flatness for 2-spheres in S4 in terms of a local π1 condition.

scholarly journals THE ALEXANDER POLYNOMIAL OF (1,1)-KNOTS

2006 ◽  
Vol 15 (09) ◽  
pp. 1119-1129 ◽  
Author(s):  
A. CATTABRIGA
Keyword(s):  
Fundamental Group ◽  
Alexander Polynomial ◽  
Lens Spaces ◽  
Cyclic Covering ◽  
Genus One

In this paper we investigate the Alexander polynomial of (1,1)-knots, which are knots lying in a 3-manifold with genus one at most, admitting a particular decomposition. More precisely, we study the connections between the Alexander polynomial and a polynomial associated to a cyclic presentation of the fundamental group of an n-fold strongly-cyclic covering branched over the knot K, which we call the n-cyclic polynomial of K. In this way, we generalize to all (1,1)-knots, with the only exception of those lying in S2×S1, a result obtained by Minkus for 2-bridge knots and extended by the author and M. Mulazzani to the case of (1,1)-knots in S3. As corollaries some properties of the Alexander polynomial of knots in S3 are extended to the case of (1,1)-knots in lens spaces.

scholarly journals Some small aspherical spaces

1973 ◽  
Vol 16 (3) ◽  
pp. 332-352 ◽  
Author(s):  
Eldon Dyer ◽  
A. T. Vasquez
Keyword(s):  
Euclidean Space ◽  
Fundamental Group ◽  
Homotopy Type ◽  
Homotopy Group ◽  
Continuous Mapping ◽  
Cell Complex ◽  
A Cell

Let Sn denote the sphere of all points in Euclidean space Rn + 1 at a distance of 1 from the origin and Dn + 1 the ball of all points in Rn + 1 at a distance not exceeding 1 from the origin The space X is said to be aspherical if for every n ≧ 2 and every continuous mapping: f: Sn → X, there exists a continuous mapping g: Dn + 1 → X with restriction to the subspace Sn equal to f. Thus, the only homotopy group of X which might be non-zero is the fundamental group τ1(X, *) ≅ G. If X is also a cell-complex, it is called a K(G, 1). If X and Y are K(G, l)'s, then they have the same homotopy type, and consequently

scholarly journals Colorings, determinants and Alexander polynomials for spatial graphs

2016 ◽  
Vol 25 (04) ◽  
pp. 1650019
Author(s):  
Blake Mellor ◽  
Terry Kong ◽  
Alec Lewald ◽  
Vadim Pigrish
Keyword(s):  
Fundamental Group ◽  
Alexander Polynomial ◽  
Metacyclic Group ◽  
Alexander Polynomials ◽  
Spatial Graph ◽  
Spatial Graphs ◽  
Integer Weight

A balanced spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to S. Kinoshita, Alexander polynomials as isotopy invariants I, Osaka Math. J. 10 (1958) 263–271.), and examine their behavior under some common operations on the graph. We use the Alexander module to define the determinant and [Formula: see text]-colorings of a balanced spatial graph, and provide examples. We show that the determinant of a spatial graph determines for which [Formula: see text] the graph is [Formula: see text]-colorable, and that a [Formula: see text]-coloring of a graph corresponds to a representation of the fundamental group of its complement into a metacyclic group [Formula: see text]. We finish by proving some properties of the Alexander polynomial.

CASSON KNOT INVARIANTS OF PERIODIC KNOTS WITH RATIONAL QUOTIENTS

2007 ◽  
Vol 16 (04) ◽  
pp. 439-460 ◽  
Author(s):  
HEE JEONG JANG ◽  
SANG YOUL LEE ◽  
MYOUNGSOO SEO
Keyword(s):  
Normal Form ◽  
Recurrence Formula ◽  
Alexander Polynomial ◽  
Knot Invariants ◽  
Knot Invariant

We give a formula for the Casson knot invariant of a p-periodic knot in S3 whose quotient link is a 2-bridge link with Conway's normal form C(2, 2n1, -2, 2n2, …, 2n2m, 2) via the integers p, n1, n2, …, n2m(p ≥ 2 and m ≥ 1). As an application, for any integers n1, n2, ≥, n2m with the same sign, we determine the Δ-unknotting number of a p-periodic knot in S3 whose quotient is a 2-bridge link C(2, 2n1, -2, 2n2, ≥, 2n2m, 2) in terms of p, n1, n2, ≥, n2m. In addition, a recurrence formula for calculating the Alexander polynomial of the 2-bridge knot with Conway's normal form C(2n1, 2n2, ≥, 2nm) via the integers n1,n2, ≥, nm is included.

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