河村 建吾

We introduce the notion of a ribbon-clasp surface-link, which is a generalization of a ribbon surface-link. We generalize the notion of a normal form on embedded surface-links to the case of immersed surface-links and prove that any immersed surface-link can be described in a normal form. It is known that an embedded surface-link is a ribbon surface-link if and only if it can be described in a symmetric normal form. We prove that an immersed surface-link is a ribbon-clasp surface-link if and only if it can be described in a symmetric normal form. We also introduce the notion of a ribbon-clasp normal form, which is a simpler version of a symmetric normal form. (C) 2017 Elsevier B.V. All rights reserved.

The Roseman moves are seven types of local modifications for surface-link diagrams in 3-space which generate ambient isotopies of surface-links in 4-space. In this paper, we focus on Roseman moves involving triple points, one of which is the famous tetrahedral move, and discuss their independence. For each diagram of any surface-link, we construct a new diagram of the same surface-link such that any sequence of Roseman moves between them must contain moves involving triple points (and the number of triple points of the two diagrams are the same). Moreover, we find a pair of diagrams of an S-2-knot such that any sequence of Roseman moves between them must involve at least one tetrahedral move.

D. 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. (C) 2015 Elsevier B.V. All rights reserved.

The clasp number c(K) of a knot K is the minimum number of clasp singularities among all clasp disks bounded by K. It is known that the genus g(K) and the unknotting number u(K) are lower bounds of the clasp number, that is, max{g(K), u(K)} <= c(K). Then it is natural to ask whether there exists a knot K such that max{g(K), u(K)} < c(K). In this paper, we prove that there exists an infinite family of prime knots such that the question above is affirmative.