On relationship between seven types of Roseman moves

Abstract

Available online xxxx MSC: 57Q45 Keywords: Surface-link 2-Link Diagram Roseman movesD. Roseman introduced seven types of local transformations of surface-link diagrams. It is known that a particular type can be realized by the other six types. There is another type that can be realized by the other six types. We show that any types except these two types cannot be realized by the other six types.

No surface-knot of genus one has triple point number two

“…By giving an Alexander number to each of the eight regions surrounding T n i , we see that it is impossible to have T n i = T n i+1 = T n i+2 . 7…”

“…For a double edge E contained We call the local deformations Roseman moves or moves. Seven types of Roseman moves in [11] can be described by seven moves shown in Figure 4 [7,19]. The deformation from the left to the right is called an R-i + move and the reverse direction is called an R-i − except R-7.…”

Independence of Roseman moves including triple points

“…In Section 2, we will generalize his result for any surface-link (including unoriented surface-links, surface-knots and S 2 -links), and prove the following. The first author [16] observed the independence of the moves of types T 1 and T 2 as local moves, and showed that there is a pair of two diagrams of the trivial S 2link with three (resp. four) components such that the pair is {T 1}-dependent (resp.…”

“…We summarize the known results below. The first and third were proved in [16] and the second was proved in [11,25].…”

On Crossing Changes for Surface-Knots

“…We will write D ∼ D to indicate that D and D present the same surface-knot. T. Yashiro [12] showed that the Roseman's seven moves can be described by six moves depicted in Figure 1 (see also [5]). We call these moves also Roseman moves.…”