On a banded link presentation of knotted surfaces

Abstract

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 1. Then we show infinitely many mutually distinct surface-knots that has an embedding with two critical points of index 1. Next we define a long flat form of a banded link for any surface-knot and show diagrammatically a long flat form of n-twist-spun (2, t)-torus knots.

Independence of the Yoshikawa eighth move and a minimal generating set of band moves

“…Examples of the links with bands of the unknotted surfaces are presented in Fig. 12, for the examples of nontrivial links with bands refer to [2]. We have the analogous (to the marked graph diagram case) moves for links with bands ([9], [7]), i.e.…”

Distances between surfaces in 4‐manifolds

“…First, we review the calculus of banded link presentations for slice discs and 2-knots set out by Jablonowski [5].…”

Minimal hard surface-unlink and classical unlink diagrams

“…An EPD code for the marked graph diagram presented in Fig. 6 [16,9,11,8], it turns out to be the minimal hard prime diagram of S 2 P 2 .…”

“…Other interpretation of the relationship between Reidemeister moves and PD codes one can find in [20]. Consider the case of ch-diagrams with zero markers, it is the case of trivial 2-links because any other base surface beside S 2 must have at least one marked vertex in any of its ch-diagram and that follows from the fact that each of its connected components satisfies the inequality: the number of marked vertices ≥ 2 − Euler characteristic (see [8]).…”

“…One can see that changing for example one vertex type in the standard torus changes its type to the unknotted sphere. Moreover, the top right image in the figure 5 on page 1231 does not represent the spun 2-knot of the trefoil because it has the trivial band (for a flat banded link representation of the spun 2-knot of the trefoil see [8]).…”