  We introduce a new combinatorial method to encode knots and links with
applications to knot invariants. Clasp diagrams defined in this paper are
combinatorial blueprints for building knot diagrams out of full twists on two
strings rather than out of crossings. We describe an equivalence relation on
clasp diagrams which produces the isotopy classes of knots as equivalence
classes. This equivalence relation is generated by local moves similar to the
Reidemeister moves.
  Clasp diagrams produce particularly simple Seifert surfaces for knots and
lead to an explicit formula for the Alexander-Conway polynomial. They are also
well-suited for the study of the Vassiliev invariants; we show that any such
invariant can be obtained via subdiagram count in the clasp diagrams.
