Surface Diagrams With at Most Two Triple Points

Abstract

In this paper, we prove that if a surface diagram of a surface-knot has at most two triple points and the lower decker set is connected, then the surface-knot group is isomorphic to the infinite cyclic group.

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

“…Since E is a b/m-or m/t-branch at T , we can apply the Roseman move R-6 − to move the branch point along E. As a result, T will be eliminated. 9 3.1 2-cancelling pair of triple points…”

“…S. Satoh showed in [16] that no 2-knot has triple point number two or three. It has been proved in [9] that if a connected orientable surface-knot has at most two triple points and the lower decker set is connected, then the fundamental group of the surface-knot is isomorphic to the infinite cyclic group. Till now, we have no examples of surface-knots with odd triple point number, even if the surface-knot is non-orientable, or disconnected.…”