ON A CALCULUS FOR 2-KNOTS AND SURFACES IN 4-SPACE

In this paper, we use knots with bands ("kwb") to define a calculus for smoothly embedded surfaces in 4-space. This approach has its roots in the classical method of moving pictures of hyperplane slices, and thus shares its simplicity and naturality. Unlike moving pictures, however, kwb are combinatorial knotlike objects in 3-space (which can, in turn, be represented by planar diagrams), and a set of three simple combinatorial moves is sufficient to connect the kwb representing isotopic surfaces. Thus we obtain a one-to-one correspondence between isotopy classes of embedded surfaces and certain kwb modulo these moves, answering a question first posed in 1994 by Katsuyuki Yoshikawa.