  We explain the notion of a grope cobordism between two knots in a 3-manifold.
Each grope cobordism has a type that can be described by a rooted unitrivalent
tree. By filtering these trees in different ways, we show how the
Goussarov-Habiro approach to finite type invariants of knots is closely related
to our notion of grope cobordism. Thus our results can be viewed as a geometric
interpretation of finite type invariants.
  An interesting refinement we study are knots modulo symmetric grope cobordism
in 3-space. On one hand this theory maps onto the usual Vassiliev theory and on
the other hand it maps onto the Cochran-Orr-Teichner filtration of the knot
concordance group, via symmetric grope cobordism in 4-space. In particular, the
graded theory contains information on finite type invariants (with degree h
terms mapping to Vassiliev degree 2^h), Blanchfield forms or S-equivalence at
h=2, Casson-Gordon invariants at h=3, and for h=4 one has the new von Neumann
signatures of a knot.
