This book offers a self-contained account of the 3-manifold
invariants arising from the original Jones polynomial. These are
the Witten-Reshetikhin-Turaev and the Turaev-Viro invariants.
Starting from the Kauffman bracket model for the Jones polynomial
and the diagrammatic Temperley-Lieb algebra, higher-order
polynomial invariants of links are constructed and combined to form
the 3-manifold invariants. The methods in this book are based on a
recoupling theory for the Temperley-Lieb algebra. This recoupling
theory is a q-deformation of the SU(2) spin networks of Roger
Penrose.
The recoupling theory is developed in a purely combinatorial and
elementary manner. Calculations are based on a reformulation of the
Kirillov-Reshetikhin shadow world, leading to expressions for all
the invariants in terms of state summations on 2-cell complexes.
Extensive tables of the invariants are included. Manifolds in these
tables are recognized by surgery presentations and by means of
3-gems (graph encoded 3-manifolds) in an approach pioneered by
Sostenes Lins. The appendices include information about gems,
examples of distinct manifolds with the same invariants, and
applications to the Turaev-Viro invariant and to the Crane-Yetter
invariant of 4-manifolds.
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