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In this talk we will introduce the notion of a cube diagram---a
three dimensional representation of a knot in R3 whose three planar knot
projections are grid diagrams. The main goal in defining cube diagrams was
to develop a data structure that describes an embedding of a knot in R3
such that (1) every link is represented by a cube diagram, (2) the data
structure is rigid enough to easily define invariants, yet (3) a limited
number of 5 inherently 3-dimensional moves are all that are necessary to
transform one cube diagram of a link into any other cube diagram of the
same link.
As an example of the usefulness of cube diagrams we will present a
homology theory constructed from cube diagrams and show that it is
equivalent to knot Floer homology, one of the most powerful known knot
invariants.
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