Links In S3 Research Articles

A generalization of Torres’ second relation

Let $L = {K_1} \cup \cdots \cup {K_\mu }$ be a tame link in ${S^3}$ of $\mu \geqslant 2$ components, and let ${L_\mu }$ be its sublink ${L_\mu } = L - {K_\mu }$. Let $H$ and ${H_\mu }$ be the abelianizations of ${\pi _1}({S^3} - L)$ and ${\pi _1}({S^3} - {L_\mu })$, respectively, and let ${t_1}, \ldots ,{t_\mu }$ (resp., ${t_1}, \ldots ,{t_{\mu - 1}}$) be the usual generators of $H$ (resp., ${H_\mu }$). If $\phi :{\mathbf {Z}}H \to {\mathbf {Z}}{H_\mu }$ is the (unique) ring homomorphism with $\phi ({t_i}) = {t_i}$ for $1 \leqslant i < \mu$, and $\phi ({t_\mu }) = 1$, then Torres’ second relation is equivalent to the statement that $\phi {E_1}(L) = (({\prod _{i < \mu }}t_i^{{l_i}}) - 1) \cdot {E_1}({L_\mu })$, where for $1 \leqslant i < \mu$, ${l_i}$ is the linking number ${l_i} = l({K_i}, {K_\mu })$. We prove that if $I{H_\mu }$ is the augmentation ideal of ${\mathbf {Z}}{H_\mu }$, then for any $k \geqslant 2$, \[ {E_{k - 1}}({L_\mu }) + \left ( {\left ( {\prod \limits _{i < \mu } {t_i^{{l_i}}} } \right ) - 1} \right )\cdot {E_k}({L_\mu }) \subseteq \phi {E_k}(L) \subseteq {E_{k - 1}}({L_\mu }) + I{H_\mu }\cdot {E_k}({L_\mu })\] and examples are given to indicate that either of these inclusions may be an equality. This theorem is used to generalize certain known properties of ${E_1}$ to the higher ideals.

How to construct all fibered knots and links

A FIBERED LINK in a 3-dimensional manifold M is a collection of disjointly imbedded circles L = LI U . . . U Lk such that M - L is the total space of a fiber bundle over S’. M - L is therefore homeomorphic to F x I with ends glued by a homeomorphism h cf F. One generally requires that the fiber F meet L nicely so that h determines the pair (M, L) and the boundary of the closure F of F is L. When M = S3 this means P is a Seifert surface for L. Alexander [l] proved that every 3-manifold contains a fibered link. In the classical case M = S3 and k = 1 it was shown by Neuwirth[lO] and Stallings[l2] that a knot L is fibered if and only if the commutator subgroup of x,(M -L) is finitely generated or, equivalently, free. In the geometric vein Murasugi[9] and Stallings[l3] have given techniques for constructing fibered knots and links: The first we will use involves “plumbing” a simple fibered link of two components to a more complicated one. A sequence of plumbings increases the genus of the fiber to any needed size. The name for this operation comes from the fact that the topological type of the new fiber is that obtained by matching the two old fibers along neighborhoods of properly imbedded arcs (Fig. 3). The second construction, called “twisting,” involves a It 1 surgery on a circle C C F which is unknotted in S3. When the Seifert linking of C is correct this surgery replaces the homeomorphism h by its composition with a Dehn twist about C. The conditions guarantee a new fibered pair in S3. The roots of this operation are in[81. This connection between Dehn twists and surgery brings in the calculus of framed links [7] and allows us to prove in § l-3 our main result: THEOREM 1. Ellery fibered link in S3 is related to the unknot by a sequence of these two operations.

On Kirby's calculus of links

(Receiued in revised form 21 July 1978) 4 FRAMED OR labelled link in S3 is a finite collection L of embedded circles in S’, each one of which is labelled by an integer. In [3] and [7] Lickorish and Wallace showed that any orientable 3-manifold can be obtained by surgery on S3 using such a link. Furthermore in [21 Kirby shows that two such manifolds are homeomorphic if and only if the links are related by a series of combinatorial moves. Thus there is a classification of orientable 3-manifolds in terms of equivalence classes of links. In this paper we present an exposition of Kirby’s theorem in a form which applies to links in a general 3-manifold and we also give a classification of non-orientable 3-manifolds by equivalence classes of links in the non-orientable S* bundle over S’ denoted S’ 5 S2. Our exposition reduces the dependence on ‘Cerf Theory’ which plays a central role in Kirby’s paper[2], and clarifies the connection between the various allowable moves. This clarification allows us to state the main classification theorem in the following simpler form: THEOREM. Orientation preserving homeomorphism classes of compact closed oriented 3-manifolds corresponds bijectively to equivalence class of labelled links in S’ where the equivalence is generated by a single move-the “Kirby move” (see §l

Surgery on links containing a cable sublink

In this paper it will be shown that there is an upper bound on the genus of any manifold obtained by Dehn surgery on a given torus link. It is also demonstrated that if L is a link in S 3 {S^3} with a cable sublink about the knot k, and surgery on at least two components of the sublink cannot be replaced by surgery on k, then the manifold resulting from surgery on L cannot be simply connected.

A family of non-invertible prime links: Corrigendum

Wilbur Whitten has pointed out to me that there exist invertible links amongst my family of allegedly non-invertible links [3], and a letter from Neville Smythe has confirmed that the error lies in the third paragraph from the bottom of page 106. Professor Smythe also suggested that the use of F-isotopy was possibly a red herring and, accordingly, I have devised a simpler proof of the existence of infinitely many non-invertible prime links in S3 (although Wilbur Whitten's paper [4] makes such a proof redundant).

A family of non-invertible prime links

The purpose of this paper is to show that there exists an infinite family of non-invertible prime 2-component links in S3. It is then noted that the existence of such a family implies the existence of a family of non-invertible wild arcs in S3, which are tame modulo an endpoint.