Higher-dimensional Knots Research Articles

On the ends of higher dimensional knot groups

An n-knot (S” in Sn+2) is said to be quasi-aspherical if Hn + i(c) = 0 where c is the universal cover of the complement of the knot. S.J. Lomonaco conjectured that all 2-knots are quasi-aspherical. We give an example of a 2-knot whose group has an infinite number of ends and we show that it is a counterexample to Lomonaco’s conjecture. We also prove: (1) every knot whose group has one or two ends is quasi-aspherical; (2) the group of every fibered knot has one or two ends; (3) the class of knots each of whose groups has one or two ends is closed under composition. The theory of ends of finitely generated groups was developed by Heinz Hopf [B] and Hans Freudenthal[6] and is based on Freudenthal’s earlier theory of the ends of a space [5]. All groups in the paper will be finitely generated unless otherwise stated. Excellent references for the theory of the ends of a group are [3, 4, 181. In [B] Hopf proved the following: (1) a group has either 0, 1, 2 or an infinite number of ends, (2) a group has 0 ends if and only if it is finite, (3) a group has two ends if and only if it has an infinite cyclic subgroup of finite index. In [20] C.T.C. Wall sharpened Hopf’s characterization of groups with 2 ends by showing that a group G has 2 ends if and only if it is one of the following two types:

Finiteness of Isotopy Classes of Certain Knots

We prove finiteness results for the isotopy classes of certain high-dimensional knots whose complement satisfies some homology conditions. In some particular cases, the isotopy class of a knot is determined by the twisted homology of its complement.

Isotopy types of knots of codimension two

In this paper the classification of n-dimensional knots in S+2, bound- ing /--connected manifolds, where 3r>n + l>6, in terms of stable homotopy theory is suggested. The problem of isotopy classification is the fundamental one of knot theory. It was solved by A. Haefliger and J. Levine for knots of codimension > 3. The first step in reduction of a classification of knots of codimension two to a homotopy problem was made by R. Lashof and J. Shaneson (6). They showed that in the class of «-dimensional knots with group Z for n > 5 there are at most two different knots having homotopy equivalent exteriors. In 1970 J. Levine (10) gave an algebraic classification of (2q — l)-dimensional knots in S2q+X bounding (q — 1)- connected manifolds. It turned out that the only invariant determining the isotopy type of such knots is the Seifert matrix considered to within 5-equivalence. Later Trotter (16) and C. Kearton (5) obtained a classification of knots, studied by J. Levine, in terms of Blanchfield pairing. C. Kearton has partially analyzed the more difficult problem of classification of 2^r-dimensional knots in S2q+2 bounding (q — l)-connected manifolds. In the present paper the classification of a wider class of higher-dimensional knots is obtained. It is the class of «-dimensional knots in S+2 bounding r-connected manifolds, where 3r > n + 1 > 6. The main result of this paper is the construction of a one-to-one correspondence between the set of isotopy types of such knots and some set described in purely homotopic terms. The plan of the paper is as follows: In §1 submanifolds of the sphere of codimension one are considered. Such a submanifold is assigned some pairing in the sense of the Spanier-Whitehead theory. This pairing describes the homotopy linking of the submanifold with its copy translated in the direction of the positive normal field. This pairing induces the usual Seifert pairing on middle dimensional homology groups and is called the homotopy Seifert pairing. The main result of § 1 is the construction of a one-to-one correspondence between the set of isotopy types of such submanifolds and the set of homotopy pairings.

Finiteness of isotopy classes of certain knots

We prove finiteness results for the isotopy classes of certain high-dimensional knots whose complement satisfies some homology conditions. In some particular cases, the isotopy class of a knot is determined by the twisted homology of its complement.

High dimensional knot groups which are not two-knot groups

This paper presents three arguments, one involving orientability, and the others Milnor duality and, respectively, the injectivity of cup product into H2 for an abelian group and free finite group actions on homotopy 3-spheres to show that there are high dimensional knot groups which are not the groups of knotted 2-spheres in S4, thus answering a question of Fox (“Some problems in knot theory”, Topology of 3-manifolds and related topics”, 168–176 (Proceedings of the University of Georgia Institute, 1961. Prentice-Hall, Englewood Cliffs, New Jersey, 1962).

Aspherical manifolds and higher-dimensional knots

Aspherical manifolds and higher-dimensional knots

Homology of regular coverings of spun CW pairs with applications to knot theory

The p-spin of a pair of CW complexes, one a subcomplex of the other, is defined. The algebraic properties of certain tensor-product chain complexes are used to calculate the homology groups of regular coverings of such spun pairs where these groups are considered as modules over the integral group ring of the group of covering transformations. In ?4, by using the free differential calculus and geometric for fundamental groups, presentations for certain homology groups are developed. In ??5 and 6 these results are used to analyze the homology and associated invariants for coverings of complements of higher-dimensional knots and torus-like embeddings in the sphere obtained by p-spinning.

The Sphericity of Higher Dimensional Knots

In 1956 CD. Papakyriakopoulos showed [5] that the complement C of a 1-sphere S1 tamely imbedded in a 3-sphere S3 is aspherical; that is, that for all i ≧ 2, πi(C) = 0. In this note we show that for n ≧ 2 the complement C of an n-sphere Sn smoothly imbedded in Sn+2 is aspherical only if the fundamental group of C is infinite cyclic. Combined with results of J. Stallings [6] or of J. Levine [3], this implies that if the complement of an Sn smoothly imbedded in Sn+2 is aspherical, n ≥ 4 , then Sn is topologically unknotted in Sn+2.

The Fundamental Ideal and π 2 of Higher Dimensional Knots

The Fundamental Ideal and π 2 of Higher Dimensional Knots

The fundamental ideal and 𝜋₂ of higher dimensional knots

Let ( S 4 , k ( S 2 ) ) ({S^4},k({S^2})) be a knot formed by spinning a polyhedral arc α \alpha about the standard 2 2 -sphere S 2 {S^2} in the 3 3 -sphere S 3 {S^3} . Then the second homotopy group of S 4 − k ( S 2 ) {S^4} - k({S^2}) as a Z π 1 Z{\pi _1} -module is isomorphic to each of the following: (1) The fundamental ideal modulo the left ideal generated by a − 1 a - 1 , where a a is the image in π 1 ( S 4 − k ( S 2 ) ) {\pi _1}({S^4} - k({S^2})) of a generator of π 1 ( S 2 − α ) {\pi _1}({S^2} - \alpha ) . (2) The first homology group of the kernel of π 1 ( S 3 − k ( S 2 ) ) → π 1 ( S 4 − k ( S 2 ) ) {\pi _1}({S^3} - k({S^2})) \to {\pi _1}({S^4} - k({S^2}))

Higher Dimensional Knots in Tubes

Let K be an n-knot in the $(n + 2)$-sphere and V a tubular neighborhood of K. Let $L’$ be an n-knot contained in a tubular neighborhood $V’$ of a trivial n-knot and L the image of $L’$ under an orientation preserving diffeomorphism of $V’$ onto V. The purpose of this paper is to show that the higher dimensional Alexander polynomial and the signature of the n-knot L are determined by those of K and $L’$.

A note on higher dimensional knots

A note on higher dimensional knots

Higher dimensional knots in tubes

Let K be an n-knot in the ( n + 2 ) (n + 2) -sphere and V a tubular neighborhood of K. Let L ′ L’ be an n-knot contained in a tubular neighborhood V ′ V’ of a trivial n-knot and L the image of L ′ L’ under an orientation preserving diffeomorphism of V ′ V’ onto V. The purpose of this paper is to show that the higher dimensional Alexander polynomial and the signature of the n-knot L are determined by those of K and L ′ L’ .

A characterization of knot polynomials

IN [7] Seifert presents a method for constructing a knot in the three-sphere S3, by imbedding a surface (with a single boundary component) in S” with prescribed self-linking characteristics. This technique was used to characterize the Alexander polynomial of a knot, in an algebraic manner. But the Alexander polynomial is only the first of a sequence of polynomials which arise as knot invariants, and it is natural to consider the problem of characterizing them algebraically. Seifert’s construction does not seem delicate enough to carry out this task.f It is the aim of this note to present a new technique for constructing knots, also based on a prescription of linking information, which can resolve the problem. The techniques to be presented here generalize to give similar characterizations of suitably defined invariants of higher-dimensional knots. This entire subject will be dealt with in a future paper; also see [4, Part II].