A characterization of quasipositive Seifert surfaces (Constructions of quasipositive knots and links, III)

This article was originally published in Topology 31 (1992). The present hyperTeXed redaction corrects a few typographical errors and updates the references.Comment: 8 pages, 5 figure

Gauge Invariant Formulation and Bosonisation of the Schwinger Model

The functional integral of the massless Schwinger model in $(1+1)$ dimensions is reduced to an integral in terms of local gauge invariant quantities. It turns out that this approach leads to a natural bosonisation scheme, yielding, in particular the famous `bosonisation rule'' and giving some deeper insight into the nature of the bosonisation phenomenon. As an application, the chiral anomaly is calculated within this formulation.Comment: LaTeX, 8 page

Algebraic functions and closed braids

This article was originally published in Topology 22 (1983). The present hyperTeXed redaction includes references to post-1983 results as Addenda, and corrects a few typographical errors. (See math.GT/0411115 for a more comprehensive overview of the subject as it appears 21 years later.)Comment: 12 pages, 2 figure

Hopf plumbing, arborescent Seifert surfaces, baskets, espaliers, and homogeneous braids

Four constructions of Seifert surfaces - Hopf plumbing, arborescent plumbing, basketry, and T-bandword handle decomposition - are described, and some interrelationships found, e.g.: arborescent Seifert surfaces are baskets; Hopf-plumbed baskets are precisely homogeneous T-bandword surfaces. A Seifert surface is Hopf-plumbed if it is a 2-disk D or if it can be constructed by plumbing a positive or negative Hopf annulus A(O,-1) or A(O,1) to a Hopf-plumbed surface along a proper arc. A Seifert surface is a basket if it is D or it can be constructed by plumbing an n-twisted unknotted annulus A(O,n) to a basket along a proper arc in D. A Seifert surface is arborescent if it is D, or it is A(O,n), or it can be constructed by plumbing A(O,n) to an arborescent Seifert surface along a transverse arc of an annulus plumband. Every arborescent Seifert surface is a basket. A tree T embedded in the complex plane C determines a set of generators for a braid group. An espalier is a tree in the closed lower halfplane with vertices on the real line R. If T is an espalier then words b in the T-generators correspond nicely to T-bandword surfaces S(b). (For example, if I is an espalier with an edge from p to p+1 for p=1,...,n-1, then the I-generators of the n-string braid group are the standard generators; when Seifert's algorithm is applied to the closed braid diagram of a word in those generators, the result is an I-bandword surface.) Theorem. For any espalier T, a T-bandword surface S(b) is a Hopf-plumbed basket iff b is homogeneous iff S(b) is a fiber surface iff S(b) is connected and incompressible. A Hopf-plumbed basket S (for instance, an arborescent fiber surface) is isotopic to a homogeneous T-bandword surface for some espalier T.Comment: AMS-LaTeX file; 23 pages, 14 figures; revised May 2000; to be published in Topology and Its Application