Joan S. Birman

New points of view in knot theory

In this article we shall give an account of certain developments in knot theory which followed upon the discovery of the Jones polynomial in 1984. The focus of our account will be recent glimmerings of understanding of the topological meaning of the new invariants. A second theme will be the central role that braid theory has played in the subject. A third will be the unifying principles provided by representations of simple Lie algebras and their universal enveloping algebras. These choices in emphasis are our own. They represent, at best, particular aspects of the far-reaching ramifications that followed the discovery of the Jones polynomial.

Braids: A Survey

Publisher Summary Crossings are suggested as they are in a picture of a highway overpass on a map. The identity braid has a canonical representation in which two strands never cross. Multiplication of braids is by juxtaposition, concatenation, isotopy, and rescaling. This chapter discusses Artins braid group, B n and its role in knot theory. The chapter illustrates ways in which structure first discovered in the braid groups generalizes to structure in Garside groups, Artin groups and surface mapping class groups. Artins braid group is naturally isomorphic to the mapping class group of an n-times punctured disc. The chapter also illustrates the topological concept of a braid and of a group of braids via the notion of a configuration space. It then outlines the new developments in the area of B n mapping, thus illustrating how to pass from diffeomorphisms to geometric braids and back again..

On the homeotopy group of a non-orientable surface

The purpose of this note is to call attention to an error in the manuscript [?]. The main result was a new proof of the main theorem in [?]. In line 1 of page 446 of [?] one finds the following remark: “This is the result obtained in [?]”. However, a comparison of the two results shows that the generator which is called d1d 2 in the notation of [?] appears in [?] but is missing from the list in [?].

On the Jones polynomial of closed 3-braids.

In [J, 2] Vaughan Jones introduced a new polynomial VL(t ) which is an invariant of the isotopy type of an oriented knot or link L c S 3. The polynomial can be computed from an arbitrary representation of L as a closed braid, i.e. from an element in one of the Artin braid groups B,, n = 1, 2, 3 . . . . . It is very powerful, distinguishing the trefoil and its mirror image, the unknot and the Kinoshita-Terasaka knot, and any two members of the infinite sequence of Whitehead links, all of which have homeomorphic complements. In an early version of [J, 2] (before the results reported here were complete) Jones had conjectured that his polynomial was injective on closed 3-braids. In this note we use trace identities in the Burau matrix representation of B 3 to construct myriad counterexamples to that conjecture. At the same time, we also disprove a second conjecture of Jones from [J, 23, that a link L is amphicheiral if VL(t)= VL(t-1). The 2-variable generalized Jones polynomial introduced in [ F Y H L M O ] also fails to distinguish any of our link pairs. Finally, our examples answer in the negative a question of Morton, who asked whether the Alexander polynomial is a complete invariant of the link obtained by adjoining the braid axis to a closed 3-braid link. In the monograph [Mu] Murasugi began a classification of closed 3-braid links, work which was extended by Hartley in [H]. From the partial results in [Mu] and [H] it seemed reasonable to conjecture that, omitting various obvious exceptional cases (links with braid index <3, composite links and torus links) the link type of a closed 3-braid is determined by its conjugacy class in Bj, up to the equivalences which correspond to orientation changes. This conjecture remains open. When we began this work we had hoped to settle it, expecting that the conjugacy class of a braid was determined by the Jones or Alexander polynomial of the associated closed braid. However, this is far from the truth. The problem of characterizing all of the 3-braids whose closures have a given Jones or Alexander or 2-variable polynomial seems to be very non-trivial and possibly even to be too complicate to have an interesting solution.

The -invariant of 3-manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented 2-manifold

Let %(ri) be the group of orientation-preserving selfhomeomorphisms of a closed oriented surface Bd U of genus n, and let 3C(n) be the subgroup of those elements which induce the identity on /f,(Bd U; Z). To each element h e. %(n) we associate a 3-manifold M(h) which is defined by a Heegaard splitting. It is shown that for each h 6 %(ri) there is a representation p of %(n) into Z/2Z such that if k £ %(n), then the u-invariant p(M(h)) is equal to the ju-invariant p(M(kh)) if and only if k 6 kernel p. Thus, properties of the 4-manifolds which a given 3-manifold bounds are related to group-theoretical structure in the group of homeomorphisms of a 2-manifold. The kernels of the homomorphisms from 3C(n) onto Z/2Z are studied and are shown to constitute a complete conjugacy class of subgroups of %(n). The class has nontrivial finite order.

Special positions for essential Tori in link complements

2-spheres, since the latter cannot be knotted. The first major attempt to understand embedded tori in link complements was a groundbreaking paper by Schubert [6]. The seminal role which is played in the topology and geometry of link complements by embedded tori was later underscored in the important work of Jaco and Shalen [3], Johansson [4] and Thurston [7], who showed that if M3 is a 3-manifold, then there is a finite collection R of essential, non-peripheral tori T,,. . .,T, in M3 such that each component of M3 split open along the tori in 0 is either Seifert-fibered or hyperbolic. Our goal in this paper is to apply the techniques of [2] to the study of essential tori in link complements. Let II be a link type in S3, with representative L. A torus T in S3-L is essential if it is incompressible, and peripheral if it is parallel to the boundary of a tubular neighborhood of L. A link type E. is simple if every essential torus in its complement is peripheral, otherwise non-simple. Satellite links (defined below) are a special case of non-simple links. To describe our results, assume that L is a closed n-braid representative of 1, with braid axis A. The axis A is unknotted, so S3-A is fibered by open discs {H,; 8 E [0,2rc]}. It will be convenient to think of A as the Z-axis in R3, and the fibers H, as half-planes at polar angle 8. Then A and the half-planes H, serve as a “coordinate system” in R3 which can be used to describe both L and T. We call our canonical embeddings types 0, 1 and k, where in the latter case k is an integer r 2. Type 0 will be familiar to most readers, but types 1 and k do not appear to have been noticed before this as general phenomena: Type 0. The torus T is the boundary of a (possibly knotted) solid torus V in S3, whose core is a closed braid with axis A. The link L is also a closed braid with respect to A, part of it being inside V and part of it (possibly empty) outside. The torus T is transverse to every fiber H, in the fibration of S3-A, and intersects each fiber in a meridian of V. It is foliated by

Studying links via closed braids IV: composite links and split links

The main result concerns changing an arbitrary closed braid representative of a split or composite link to one which is obviously recognizable as being split or composite. Exchange moves are introduced; they change the conjugacy class of a closed braid without changing its link type or its braid index. A closed braid representative of a composite (respectively split) link is composite (split) if there is a 2-sphere which realizes the connected sum decomposition (splitting) and meets the braid axis in 2 points. It is proved that exchange moves are the only obstruction to representing composite or split links by composite or split closed braids. A special version of these theorems holds for 3 and 4 braids, answering a question of H. Morton. As an immediate Corollary, it follows that braid index is additive (resp. additive minus 1) under disjoint union (resp. connected sum).

An inverse function theorem for free groups

AMrnACr. Let Fi be a free group of rank n with free basis xl, * **x,,. Let fyi, , ykJ be a set of k<n elements of F,,, where each yi is represented by a word Yi(xL, , x,,) in the generators xi. Let aylfax, denote the free derivative of yi with respect to xi, and let Jk,n= IIayilax,jI denote the k xn Jacobian matrix. THEoREM. If k=n, the set {yl, * * * y,,} generates F,, if and only if J,,,, has a right inverse. If k<n, the set {yiL, , yk} may be extended to a set of elements which generate F, only if Jkn has a right inverse. Several applications are given.

Stabilization in the braid groups II: Transversal simplicity of knots

The main result of this paper is a negative answer to the question: are all transversal knot types transversally simple? An explicit infinite family of examples is given of closed 3-braids that define transversal knot types that are not transversally simple. The method of proof is topological and indirect.

Studying links via closed braids II: On a theorem of Bennequin

Abstract Links which are closed 3-braids admit very special types of spanning surfaces of maximal Euler characteristic. These surfaces are decribed naturally by words in cyclically symmetric elementary braids which generate the group B 3 .