Encoding knots by clasp diagrams

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.