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This book is meant to serve either as a textbook for an
interdisciplinary course in Mathematics of Design, or as a trade
book for designers. It will also be of interest for people
interested in recreational mathematics showing the connection
between mathematics and design. Topics from the book can also be
adapted for use in pre-college mathematics. Each chapter will
provide the user with ideas that can be incorporated in a design.
Background materials will be provided to show the reader the
mathematical principles that lie behind the designs.
This volume consists primarily of survey papers that evolved from
the lectures given in the school portion of the meeting and
selected papers from the conference.Knot theory is a very special
topological subject: the classification of embeddings of a circle
or collection of circles into three-dimensional space. This is a
classical topological problem and a special case of the general
placement problem: Understanding the embeddings of a space X in
another space Y. There have been exciting new developments in the
area of knot theory and 3-manifold topology in the last 25 years.
From the Jones, Homflypt and Kauffman polynomials, quantum
invariants of 3-manifolds, through, Vassiliev invariants,
topological quantum field theories, to relations with gauge theory
type invariants in 4-dimensional topology.More recently, Khovanov
introduced link homology as a generalization of the Jones
polynomial to homology of chain complexes and Ozsvath and Szabo
developed Heegaard-Floer homology, that lifts the Alexander
polynomial. These two significantly different theories are closely
related and the dependencies are the object of intensive study.
These ideas mark the beginning of a new era in knot theory that
includes relationships with four-dimensional problems and the
creation of new forms of algebraic topology relevant to knot
theory. The theory of skein modules is an older development also
having its roots in Jones discovery. Another significant and
related development is the theory of virtual knots originated
independently by Kauffman and by Goussarov Polyak and Viro in the
'90s. All these topics and their relationships are the subject of
the survey papers in this book.It is a remarkable fact that knot
theory, while very special in its problematic form, involves ideas
and techniques that involve and inform much of mathematics and
theoretical physics. The subject has significant applications and
relations with biology, physics, combinatorics, algebra and the
theory of computation. The summer school on which this book is
based contained excellent lectures on the many aspects of
applications of knot theory. This book gives an in-depth survey of
the state of the art of present day knot theory and its
applications.
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