It is known that there is no 2-knot with triple point number two. The present paper shows that there is no surface-knot of genus one with triple point number two. In order to prove the result, we use Roseman moves and the algebraic intersection number of simple closed curves in the double decker set.

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In this paper, we discuss the crossing change operation along exchangeable double curves of a surface-knot diagram. We show that under certain condition, a finite sequence of Roseman moves preserves the property of those exchangeable double curves. As an application for this result, we also define a numerical invariant for a set of surface-knots called du-exchangeable set.

In this paper, we show that there is no surface-knot of genus one with triple point number invariant equal to three. Double point curves of surface-knot diagramsLet ∆ be a surface-knot diagram of a surface-knot F . By connecting double edges which are in opposition to each other at each triple point of ∆, the singularity set of a projection is regarded as a union of curves (circles and arc components) immersed in 3-space. We call such an oriented curve a double point curve. A double point curve with no triple points is called trivial. Lemma 2.1 ([8]). The number of triple points along each double point circle is even.

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