  Let $f(\bfz,\bar\bfz)$ be a mixed strongly polar homogeneous polynomial of
$3$ variables $\bfz=(z_1,z_2, z_3)$. It defines a Riemann surface
$V:=\{[\bfz]\in \BP^{2}\,|\,f(\bfz,\bar\bfz)=0 \}$ in the complex projective
space $\BP^{2}$. We will show that for an arbitrary given $g\ge 0$, there
exists a mixed polar homogeneous polynomial with polar degree 1 which defines a
projective surface of genus $g$. For the construction, we introduce a new type
of weighted homogeneous polynomials which we call {\em polar weighted
homogeneous polynomials of twisted join type}.
