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Books > Science & Mathematics > Mathematics > Topology > Algebraic topology
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Gauss Diagram Invariants for Knots and Links (Hardcover, 2001 ed.)
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Gauss Diagram Invariants for Knots and Links (Hardcover, 2001 ed.)
Series: Mathematics and Its Applications, 532
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Gauss diagram invariants are isotopy invariants of oriented knots
in- manifolds which are the product of a (not necessarily
orientable) surface with an oriented line. The invariants are
defined in a combinatorial way using knot diagrams, and they take
values in free abelian groups generated by the first homology group
of the surface or by the set of free homotopy classes of loops in
the surface. There are three main results: 1. The construction of
invariants of finite type for arbitrary knots in non orientable
3-manifolds. These invariants can distinguish homotopic knots with
homeomorphic complements. 2. Specific invariants of degree 3 for
knots in the solid torus. These invariants cannot be generalized
for knots in handlebodies of higher genus, in contrast to
invariants coming from the theory of skein modules. 2 3. We
introduce a special class of knots called global knots, in F x lR
and we construct new isotopy invariants, called T-invariants, for
global knots. Some T-invariants (but not all !) are of finite type
but they cannot be extracted from the generalized Kontsevich
integral, which is consequently not the universal invariant of
finite type for the restricted class of global knots. We prove that
T-invariants separate all global knots of a certain type. 3 As a
corollary we prove that certain links in 5 are not invertible
without making any use of the link group! Introduction and
announcement This work is an introduction into the world of Gauss
diagram invariants.
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