Akiko Shima

The 2-twist-spun trefoil has the triple point number four

The triple point number of an embedded surface in 4-space is the minimal number of the triple points on all the projection images into 3-space. We show that the 2-twist-spun trefoil has the triple point number four.

Two filtrations of ribbon 2-knots

Abstract We have constructed a ‘Vassiliev-like’ filtration on the free abelian group generated by the set of ribbon 2-knots in 4-space in two ways: one is from a ribbon 2-disk, and the other from a projection of a ribbon 2-knot onto a generic 3-space whose singular set consists of only double points. Each filtration determines a notion of finite type invariants for ribbon 2-knots. We prove that the two filtrations are the same, and thus, the two finite type invariants are coincident.

THE CLOSURES OF SURFACE BRAIDS OBTAINED FROM MINIMAL n-CHARTS WITH FOUR WHITE VERTICES

In this paper, we investigate surface braids obtained from minimal charts with exactly four white vertices.

COLORINGS AND ALEXANDER POLYNOMIALS FOR RIBBON 2-KNOTS

Let K be a ribbon 2-knot. We show that for any ideal J of the Laurent polynomial ring Λ, if Alexander polynomail of K is trivial, i.e., ΔK(t) = 1, then all colorings of K on Λ/J are trivial.

Gambits in charts

In this paper, we develop a method to change the label of a ring in a chart by C-moves and a stabilization where a ring is a simple closed curve consisting of edges of the same label which does not contain any white vertices.

On charts with two crossings II

Let 0 be a chart with at most two crossings. In this paper, we show th at if 0 is a 2-minimal generalizedn-chart with n 5, then0 contains at least 4 n 10 black vertices. And we show that if the closure of the surface braid represented by 0 is a disjoint union of spheres, then 0 is a ribbon chart. Hence the closure is a ribbon surface.

Separating a Chart

In this paper, we shall show a condition for that a chart is C-move equivalent to the product of two charts, the union of two charts

Properties of minimal charts and their applications I

\Gamma^*

Knotted Klein bottles with only double points

and

MINIMAL n-CHARTS WITH FOUR WHITE VERTICES

\Gamma^{**}