arXiv subjects
Publications and source records attributed to Joan S. Birman.
In this paper we use ideas introduced earlier by Moody, Long, Long-Paton, and Bigelow to prove the theorem of the title, that the Burau representation of the classical braid group is faithful for n = 4. An immediate corollary is that the Jones representation of the braid group is also faithful for n = 4.
In this article the discovery of the Jones Polynomial will be discussed, emphasizing the way in which it illustrated the remarkable unity between distinct parts of Mathematics, each with its own language, but initially without a dictionary.
In this work, we study the cellular decomposition of $S$ induced by a filling pair of curves $v$ and $w$, $Dec_{v,w}(S) = S - (v \cup w)$, and its connection to the distance function $d(v,w)$ in the curve graph of a closed orientable surface $S$ of genus $g$. Efficient geodesics were introduced by the first author in joint work with Margalit and Menasco in 2016, giving an algorithm that begins with a pair of non-separating filling curves that determine vertices $(v,w)$ in the curve graph of a closed orientable surface $S$ and computing from them a finite set of efficient geodesics. We extend the tools of efficient geodesics to study the relationship between distance $d(v,w)$, intersection number $i(v,w)$, and $Dec_{v,w}(S)$. The main result is the development and analysis of particular configurations of rectangles in $Dec_{v,w}(S)$ called spirals. We are able to show that, in some special cases, the efficient geodesic algorithm can be used to build an algorithm that reduces $i(v,w)$ while preserving $d(v,w)$. At the end of the paper, we note a connection of our work to the notion of extending geodesics.
It is proved that the curve graph $C^1(Σ)$ of a surface $Σ_{g,n}$ has a local pathology that had not been identified as such: there are vertices $α,β$ in $C^1(Σ)$ such that $β$ is a dead end of every geodesic joining $α$ to $β$. It also has double dead-ends. Every dead end has depth 1.
It has been known since 1981 that if one fixes an orientable surface $S$ of genus $g$, then there is a real number $λ_{min,g} > 1$ that is the dilatation of a pA diffeomorphism of $S$, and every other pA diffeomorphism of $S$ has dilatation $\geq λ_{min,g}$. We will show how a little-known theorem about digraphs gives some insight into $λ_{min,g}$.
It has been known since 1981 that if one fixes an orientable surface $S$ of genus $g$, then there is a real number $λ_{min,g} > 1$ that is the dilatation of a pA diffeomorphism of $S$, and every other pA diffeomorphism of $S$ has dilatation $\geq λ_{min,g}$. We will show how a little-known theorem about digraphs gives some insight into $λ_{min,g}$.
Knots and links which are closed 3-braids are a very special class. Like 2-bridge knots and links, they are simple enough to admit a complete classification. At the same time they are rich enough to serve as a source of examples on which, with luck, a researcher may be able to test various conjectures. The goal of this review article is to gather together, in one place, some of the tools that are special to knots and links of braid index 3, in a form that could be useful for those who have a need to calculate, and need to know precisely all the exceptional cases. We also use it as an opportunity to review what is known and suggest some open questions.
Let $f$ be the gluing map of a Heegaard splitting of a 3-manifold $W$. The goal of this paper is to determine the information about $W$ contained in the image of $f$ under the symplectic representation of the mapping class group. We prove three main results. First, we show that the first homology group of the three manifold together with Seifert's linking form provides a complete set of stable invariants. Second, we give a complete, computable set of invariants for these linking forms. Third, we show that a slight augmentation of Birman's determinantal invariant for a Heegaard splitting gives a complete set of unstable invariants.
The contents of this 6-page paper have been subsumed into the 13-page paper, "A note on closed 3-braids", arXiv:0802.1072 [math.GT]. This paper is correct, but contains less information than the new one. The topological classification of knots that are closed braids is shown to lead to a classification theorem for transversal knots that are closed 3-braids. A list is given of all low crossing examples of transversally non-simple knots that are closed 3-braids.
Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus $g$ with one boundary component to $\wedge^3 H$, the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping class group to ${1/2} \wedge^3 H \semi \sp(H)$. This Johnson-Morita homomorphism is not surjective, but its image is finite index in ${1/2} \wedge^3 H \semi \sp(H)$. Here we give a description of the exact image of Morita's homomorphism. Further, we compute the image of the handlebody subgroup of the mapping class group under the same map.
An element in Artin's braid group B_n is said to be periodic if some power of it lies in the center of B_n. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in B_n are exponential in the braid index n for the special case of periodic braids. We overcome this difficulty by putting to work several known isomorphisms between Garside structures in the braid group B_n and other Garside groups. This allows us to obtain a polynomial solution to the original problem in the spirit of the previously known algorithms. This paper is the third in a series of papers by the same authors about the conjugacy problem in Garside groups. They have a unified goal: the development of a polynomial algorithm for the conjugacy decision and search problems in B_n, which generalizes to other Garside groups whenever possible. It is our hope that the methods introduced here will allow the generalization of the results in this paper to all Artin-Tits groups of spherical type.
Withdrawn and replaced by two related manuscripts: (1) "Stabilization in the braid groups I:MTWS", published in Geometry and Topology Volume 10 (2006), 413-540, arXiv:math.GT/0310279, and (2) "Stabilization in the braid groups II: Transversal simplicity of knots", Geometry and Topology Volume 10 (2006), to appear, arXiv:math,GT/0310280.
This article is the author's contribution to the volume "Problems on mapping class groups and related topics" which will be published in December 2006, with Benson Farb as Editor. Various individuals were invited by Farb to submit open problems which seemed interesting to them about surface mapping class groups. We singled out a particular recent development, Hempel's concept of the `Heegaard distance' of an element in the mapping class group, and used it to describe a number of open questions which relate to the correspondence between Heegaard surfaces in 3-manifolds and the mapping class group of a 2-manifold.
This paper is the second in a series in which the authors study the conjugacy decision problem (CDP) and the conjugacy search problem (CSP) in Garside groups. The ultra summit set USS(X) of an element X in a Garside group G is a finite set of elements in G, introduced by the second author, which is a complete invariant of the conjugacy class of X in G. A fundamental question, if one wishes to find bounds on the size of USS(X), is to understand its structure. In this paper we introduce two new operations on elements of USS(X), called `partial cycling' and `partial twisted decycling', and prove that if Y and Z belong to USS(X), then Y and Z are related by sequences of partial cyclings and partial twisted decyclings. These operations are a concrete way to understand the minimal simple elements which result from the convexity theorem in the mentioned paper by the second author. Using partial cycling and partial twisted decycling, we investigate the structure of a directed graph Γ_X which is determined by USS(X), and show that Γ_X can be decomposed into `black' and `grey' subgraphs. There are applications which relate to the program, outlined in the first paper in this series, for finding a polynomial solution to the CDP/CSP in the case of braids. A different application is to give a new algorithm for solving the CDP/CSP in Garside groups which is faster than all other known algorithms, even though its theoretical complexity is the same as that given by the second author. There are also applications to the theory of reductive groups.
In this paper a relation between iterated cyclings and iterated powers of elements in a Garside group is shown. This yields a characterization of elements in a Garside group having a rigid power, where 'rigid' means that the left normal form changes only in the obvious way under cycling and decycling. It is also shown that, given X in a Garside group, if some power X^m is conjugate to a rigid element, then m can be bounded above by ||Δ||^3. In the particular case of braid groups, this implies that a pseudo-Anosov braid has a small power whose ultra summit set consists of rigid elements. This solves one of the problems in the way of a polynomial solution to the conjugacy decision problem (CDP) and the conjugacy search problem (CSP) in braid groups. In addition to proving the rigidity theorem, it will be shown how this paper fits into the authors' program for finding a polynomial algorithm to the CDP/CSP, and what remains to be done.
This article is about Artin's braid group and its role in knot theory. We set ourselves two goals: (i) to provide enough of the essential background so that our review would be accessible to graduate students, and (ii) to focus on those parts of the subject in which major progress was made, or interesting new proofs of known results were discovered, during the past 20 years. A central theme that we try to develop is to show ways in which structure first discovered in the braid groups generalizes to structure in Garside groups, Artin groups and surface mapping class groups. However, the literature is extensive, and for reasons of space our coverage necessarily omits many very interesting developments. Open problems are noted and so-labelled, as we encounter them.
The purpose of this erratum is to fill a gap in the proof of the `Composite Braid Theorem' in the manuscript "Studying Links Via Closed Braids IV: Composite Links and Split Links (SLVCB-IV)", Inventiones Math, \{bf 102\} Fasc. 1 (1990), 115-139. The statement of the theorem is unchanged. The gap occurs on page 135, lines $13^-$ to $11^-$, where we fail to consider the case: $V_2 = 4, V_4 > 0, V_j=0$ if $j not= 2,4,$ and all 4 vertices of valence 2 are bad. At the end of this Erratum we make some brief remarks on the literature, as it has evolved during the 14 years between the publication of SLVCB-IV and the submission of this Erratum.