Ralph H. Fox

Metacyclic invariants of knots and links

To each representation ρ on a transitive permutation group P of the group G = π(S – k) of an (ordered and oriented) link k = k1 ∪ k2 ∪ … ∪ kμ in the oriented 3-sphere S there is associated an oriented open 3-manifold M = Mρ(k), the covering space of S – k that belongs to ρ. The points 01, 02, … that lie over the base point o may be indexed in such a way that the elements g of G into which the paths from oi to oj project are represented by the permutations gρ of the form , and this property characterizes M. Of course M does not depend on the actual indices assigned to the points o 1, o 2, … but only on the equivalence class of ρ, where two representations ρ of G onto P and ρ′ of G onto P′ are equivalent when there is an inner automorphism θ of some symmetric group in which both P and P′ are contained which is such that ρ′ = θρ.

On the Imbedding of Polyhedra in 3-Space

The object of our investigation is spherical 3-space S and its polyhedral subsets. The sets which come into consideration are all to be polyhedral (or polygonal) even when this is not explicitly stated; all homeomorphisms considered are to be semi-linear. Given in S any finite (positive) number of non-intersecting closed surfaces, the closure of any component of the residual space will be called a connected elementary figure. An elementary figure [1] is the union of a finite collection of non-intersecting connected elementary figures. It is known [2] that any polyhedron P in S which is not the whole space has a closed neighborhood, called a regular neighborhood, which is an elementary figure and of which P is a deformation retract. Thus in any investigation of properties of the homotopy types [3] of polyhedra in S one may restrict oneself to consideration of connected elementary figures. A further simplification is effected by theorem (1) below, whose proof is our main objective. We shall often have to deal with the closure of the complement of an elementary figure; this set, which is itself an elementary figure, will be designated as the closed complement, or more simply as the complement, of the given elementary figure. A regular neighborhood of a finite (but not necessarily connected) linear graph will be called a tubular figure. The combinatorial type, and hence the topological type, of such a figure is uniquely determined by the given graph: a connected graph of Euler characteristic 1 p gives rise to a connected tubular figure whose boundary is an orientable surface of genus p and which is of the topological type of the Cartesian product of a 1-cell with a closed simplyor multiply-connected region of the plane. An example of an elementary figure which is not tubular may be obtained by boring through a 3-cell a system of tunnels, which may be knotted or linked and need not have entrances or exits. That this is a description. of the general non-tubular elementary figure is the content of theorem (1). MAIN THEOREM (1) Every connected elementary figure in S is homeomorphic to the closed complement of some tubular figure in S. An immediate consequence of this theorem is that any connected polyhedron in S belongs to the homotopy type of the residual space of some finite graph in S. This shows that questions concerning the homotopy type of polyhedra that can be imbedded in Euclidean 3-space may be studied by the methods of knot theory. Following the proof of (1) several results of this nature are derived. We begin by applying a suitable modification of a method, used by Alexander [4], to obtain a proof of the following lemma: (2) Let Ui, 022 )... * Ur be a system of non-intersecting surfaces in S and suppose that these surfaces are not all of genus zero. Then there exists a 2-cell C whose interior C C is disjoint to a = ai + . . . + Urn and whose boundary curve C is on u (i.e. is a subset of a) but does not bound any 2-cell on a.

An ideal class invariant of knots

In this note the row and column ideal class invariants of a matrix (cf. [3]) are applied to the Alexander matrix of a knot to give an invariant for knots. We give an example of the use of this invariant to distinguish a pair of knots which cannot be distinguished by their elementary ideals, torsion numbers, or linking invariants.I 1. Let M be an m X n matrix over a commutative integral domain R. For any integer k consider the kth compound matrix M(k), the

On fibre spaces. II

This paper is primarily concerned with fibre mappings into an absolute neighborhood retract. Theorem 3 is a converse of the covering homotopy theorem; it characterizes fibre mappings (into a compact ANR) as mappings for which the covering homotopy theorem holds. Theorem 4 is Borsuks fibre theorem; the proof which I present here is new. It seems to me that this theorem is a promising tool in function-space theory. Also I think that it furnishes conclusive justification for the generality of the Hurewicz-Steenrod definition of a fibre space. In fact, a fibre space of the type constructed by Borsuks theorem almost never has a compact base space and almost never has its fibres of the same topological type. The common denominator of the proofs of Theorems 3 and 4 is a property which I call local equiconnectivity. Local equiconnectivity is a strengthened form of local contractibility and a weakened form of the absolute neighborhood retract property (Theorems 1 and 2). Definitions and notations are those of FS. I. Let A be the diagonal subset ]C&e#(^ &) °f 5 X 5 . I shall call the space B locally equiconnected (or, to be specific, (U, F)-equiconnected) if there are neighborhoods U and F of A and a homotopy X in B between the two projections of U which does not move the points of A and which is uniform with respect to V. Precisely: (1) X*(eo, fa) is defined for all (&0, fa) £ Uy (2) Xo(6o, &i)=*o,

Calculation of Fundamental Groups

It was remarked in Chapter II that a rigorous calculation of the fundamental group of a space X is rarely just a straightforward application of the definition of π(X). At this point the collection of topological spaces whose fundamental groups the reader can be expected to know (as a result of the theory so far developed in this book) consists of spaces topologically equivalent to the circle or to a convex set. This is not a very wide range, and the purpose of this chapter is to do something about increasing it. The techniques we shall consider are aimed in two directions. The first is concerned with what we may call spaces of the same shape. Figures 14, 15, and 16 are examples of the sort of thing we have in mind. From an understanding of the fundamental group as formed from the set of classes of equivalent loops based at a point, it is geometrically apparent that the spaces shown in Figure 14 below have the same, or isomorphic, fundamental groups.

The Knot Polynomials

The underlying knot-theoretic structure developed in this book is a chain of successively weaker invariants of knot type. The sequence of knot polynomials, to which this chapter is devoted, is the last in the chain

Presentation of Groups

\begin{gathered} \quad \quad \quad \quad knot\,type\,of\,K \hfill \\ \quad \quad \quad \quad \quad \quad \downarrow \hfill \\ presentation\,type\;of\,\pi \left( {{R^3} - K} \right) \hfill \\ \quad \quad \quad \quad \quad \quad \downarrow \hfill \\ sequence\,of\,elementary\,ideals \hfill \\ \quad \quad \quad \quad \quad \;\;\; \downarrow \hfill \\ sequence\,of\,knot\,polynomials \hfill \\ \end{gathered}

The Fundamental Group

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