On a banded link presentation of knotted surfaces

Abstract

We will discuss a method for visual presentation of knotted surfaces in the four space, by examining a number and a position of its Morse’s critical points. Using this method, we will investigate surface-knot with one critical point of index [Formula: see text]. Then we show infinitely many mutually distinct surface-knots that have an embedding with two critical points of index [Formula: see text]. Next we define a long flat form of a banded link for any surface-knot and show diagrammatically a long flat form of [Formula: see text]-twist-spun [Formula: see text]-torus knots.