Teruo Nagase

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.

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

On the signature of involutions on an oriented closed 3-manifold

\Gamma^*

Properties of minimal charts and their applications I

and

MINIMAL n-CHARTS WITH FOUR WHITE VERTICES

\Gamma^{**}

On surface braids of index four with at most two crossings

which are contained in disks

Any chart with at most one crossing is a ribbon chart

D^*

The closure of a surface braid represented by a 4-chart with at most one crossing is a ribbon surface

and