On flat braidzel surfaces for links

Rudolph introduced the notion of braidzel surfaces as a generalization of pretzel surfaces, and Nakamura showed that any oriented link has a braidzel surface. In this paper, we introduce the notion of flat braidzel surfaces as a special kind of braidzel surfaces, and show that any oriented link has a flat braidzel surface. We also introduce and study a new integral invariant of links, named the flat braidzel genus, with respect to their flat braidzel surfaces. Moreover, we give a way to calculate the number of components, the distance from proper links, the Arf invariant, and a Seifert matrix of a given link through the flat braidzel notation.