7,436 research outputs found
Erratum: Studying Links via Closed Braids IV: Composite Links and Split Links
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 to , where we fail to consider the case: if 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.Comment: 6 pages, 4 figures. This is an Erratum to "Studying Links Vai Closed
Braids IV: Composite Links and Split Links", Inventiones Math., 102 Facs. 1
(190), 115-13
On 4-fold covering moves
We prove the existence of a finite set of moves sufficient to relate any two
representations of the same 3-manifold as a 4-fold simple branched covering of
S^3. We also prove a stabilization result: after adding a fifth trivial sheet
two local moves suffice. These results are analogous to results of Piergallini
in degree 3 and can be viewed as a second step in a program to establish
similar results for arbitrary degree coverings of S^3.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-5.abs.html Version 2:
correction added on page 13
Eigenvalue bounds in the gaps of Schrodinger operators and Jacobi matrices
We consider where is selfadjoint with a gap in its
spectrum and is (relatively) compact. We prove a general result allowing
of indefinite sign and apply it to obtain a bound for
perturbations of suitable periodic Schrodinger operators and a (not
quite)Lieb-Thirring bound for perturbations of algebro-geometric almost
periodic Jacobi matrices
An elementary approach to the mapping class group of a surface
We consider an oriented surface S and a cellular complex X of curves on S,
defined by Hatcher and Thurston in 1980. We prove by elementary means, without
Cerf theory, that the complex X is connected and simply connected. From this we
derive an explicit simple presentation of the mapping class group of S,
following the ideas of Hatcher-Thurston and Harer.Comment: 62 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTVol3/paper17.abs.htm
Spectral approach to homogenization of an elliptic operator periodic in some directions
The operator is considered in
, where , are periodic in
with period 1, bounded and positive definite. Let function
be bounded, positive definite and periodic in with
period 1. Let . The
behavior of the operator as
is studied. It is proved that the operator tends to in the operator norm in
. Here is the effective operator whose
coefficients depend only on , is the mean value of in
. A sharp order estimate for the norm of the difference
is obtained.
The result is applied to homogenization of the Schr\"odinger operator with a
singular potential periodic in one direction.Comment: 3
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.Comment: This is the version published by Geometry & Topology on 4 October
2006. Part I (arXiv:math/0310279) is also published in this volum
Lagrangian mapping class groups from a group homological point of view
We focus on two kinds of infinite index subgroups of the mapping class group
of a surface associated with a Lagrangian submodule of the first homology of a
surface. These subgroups, called Lagrangian mapping class groups, are known to
play important roles in the interaction between the mapping class group and
finite-type invariants of 3-manifolds. In this paper, we discuss these groups
from a group (co)homological point of view. The results include the
determination of their abelianizations, lower bounds of the second homology and
remarks on the (co)homology of higher degrees. As a by-product of this
investigation, we determine the second homology of the mapping class group of a
surface of genus 3.Comment: 20 pages. The proof of Lemma 4.2 is corrected. To appear in Algebraic
& Geometric Topolog
Homogenization of the elliptic Dirichlet problem: operator error estimates in
Let be a bounded domain of class . In
the Hilbert space , we consider a matrix
elliptic second order differential operator with
the Dirichlet boundary condition. Here is the small parameter.
The coefficients of the operator are periodic and depend on
. A sharp order operator error estimate
is obtained. Here is the effective
operator with constant coefficients and with the Dirichlet boundary condition.Comment: 13 page
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