Braids: A Survey

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.Comment: Final version, revised to take account of the comments of readers. A review article, to appear in the Handbook of Knot Theory, edited by W. Menasco and M. Thistlethwaite. 91 pages, 24 figure

On the spectra of mapping classes and the 4-genera of positive knots

Roughly, this thesis can be divided into three parts. In the first part, we study the Galois conjugates of the dilatation of pseudo-Anosov mapping classes. In particular, for a product of two multitwists, we show that all Galois conjugates are either real and positive or contained in the unit circle and the positive real axis, depending on whether the products are of opposite or of the same sign. Furthermore, for each closed orientable surface, we determine the minimal dilatation among pseudo-Anosov mapping classes arising from Penner's construction. In the second part, we consider the Alexander polynomial and the signature function of links. For a Murasugi sum of two Seifert surfaces with symmetric, definite Seifert form, we show that all zeroes of the Alexander polynomial are either real and positive or contained in unit circle and the negative real axis, depending on whether the Seifert forms are definite of opposite or the same sign. Furthermore, we prove that the signature function of a Murasugi sum of two Seifert surfaces with symmetric, definite Seifert form is monotonic. We also show that the signature of a positive arborescent Hopf plumbing is greater than or equal to two thirds of the first Betti number. In the third part, we study the topological four-genus of positive braid knots. We show that the difference of the ordinary Seifert genus and the topological four-genus grows at least linearly with the positive braid index. In particular, we show that the positive braid knots for which the topological four-genus equals the ordinary Seifert genus are exactly the positive braid knots with maximal signature invariant