Roger Fenn

RACKS AND LINKS IN CODIMENSION TWO

A rack, which is the algebraic distillation of two of the Reidemeister moves, is a set with a binary operation such that right multiplication is an automorphism. Any codimension two link has a fundamental rack which contains more information than the fundamental group. Racks provide an elegant and complete algebraic framework in which to study links and knots in 3–manifolds, and also for the 3–manifolds themselves. Racks have been studied by several previous authors and have been called a variety of names. In this first paper of a series we consolidate the algebra of racks and show that the fundamental rack is a complete invariant for irreducible framed links in a 3–manifold and for the 3–manifold itself. We give some examples of computable link invariants derived from the fundamental rack and explain the connection of the theory of racks with that of braids.

Trunks and classifying spaces

Trunks are objects loosely analogous to categories. Like a category, a trunk has vertices and edges (analogous to objects and morphisms), but instead of composition (which can be regarded as given by preferred triangles of morphisms) it has preferred squares of edges. A trunk has a natural cubical nerve, analogous to the simplicial nerve of a category. The classifying space of the trunk is the realisation of this nerve. Trunks are important in the theory of racks [8]. A rackX gives rise to a trunkT (X) which has a single vertex and the setX as set of edges. Therack spaceBX ofX is the realisation of the nerveNT (X) ofT(X). The connection between the nerve of a trunk and the usual (cubical) nerve of a category determines in particular a natural mapBX ↦BAs(X) whereBAs(X) is the classifying space of the associated group ofX. There is an extension to give a classifying space for an augmented rack, which has a natural map to the loop space of the Brown-Higgins classifying space of the associated crossed module [8, Section 2] and [3].The theory can be used to define invariants of knots and links since any invariant of the rack space of the fundamental rack of a knot or link is ipso facto an invariant of the knot or link.

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

The braid-permutation group

Abstract We consider the subgroup of the automorphism group of the free group generated by the braid group and the permutation group. This is proved to be the same as the subgroup of automorphisms of permutation-conjugacy type and is represented by generalised braids (braids in which some crossings are allowed to be “welded”). As a consequence of this representation there is a finite presentation which shows the close connection with both the classical braid and permutation groups. The group is isomorphic to the automorphism group of the free quandle and closely related to the automorphism group of the free rack. These automorphism groups are connected with invariants of classical knots and links in the 3-sphere.

The rack space

The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space and the classifying bundle is the first James bundle. We investigate the algebraic topology of this classifying space and report on calculations given elsewhere. Apart from de. ning many new knot and link invariants (including generalised James-Hopf invariants), the classification theorem has some unexpected applications. We give a combinatorial interpretation for pi(2) of a complex which can be used for calculations and some new interpretations of the higher homotopy groups of the 3-sphere. We also give a cobordism classification of virtual links.

QUATERNIONIC INVARIANTS OF VIRTUAL KNOTS AND LINKS

In this paper we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 2 x 2 matrices with entries in a possibly non-commutative ring, for example the quaternions. These polynomials are sufficiently powerful to distinguish the Kishino knot from any classical knot, including the unknot

An Introduction to Species and the Rack Space

Racks were introduced in [FR]. In this paper we define a natural category like object, called a species.* A particularly important species is associated with a rack. A species has a nerve, analogous to the nerve of a category, and the nerve of the rack species yields a space associated to the rack which classifies link diagrams labelled by the rack. We compute the second homotopy group of this space in the case of a classical rack. This is a free abelian group of rank the number of non-trivial maximal irreducible sublinks of the link.

The singular braid monoid embeds in a group

We prove that the singular braid monoid of [2] and [5] embeds in a group. This group has a geometric interpretation as singular braids with two type of singularities which cancel.

Weyl algebras and knots

Abstract In this paper we put forward results on the invariant F -module of a virtual knot investigated by the first named author where F is the algebra with two invertible generators A , B and one relation A − 1 B − 1 A B − B − 1 A B = B A − 1 B − 1 A − A . For flat knots and links the two sides of the relation equation are put equal to unity and the algebra becomes the Weyl algebra. If this is perturbed and the two sides of the relation equation are put equal to a general element, q , of the ground ring, then the resulting module lays claim to be the correct generalization of the Alexander module. Many finite dimensional representations are given together with calculations.

BIQUANDLES OF SMALL SIZE AND SOME INVARIANTS OF VIRTUAL AND WELDED KNOTS

In this paper we give the results of a computer search for biracks of small size and we give various interpretations of these findings. The list includes biquandles, racks and quandles together with new invariants of welded knots and examples of welded knots which are shown to be non-trivial by the new invariants. These can be used to answer various questions concerning virtual and welded knots. As an application we reprove the result that the Burau map from braids to matrices is non injective and give an example of a non-trivial virtual (welded) knot which cannot be distinguished from the unknot by any linear biquandles. (This is a revised version of an earlier paper of the same name)