  A twisted torus knot is a knot obtained from a torus knot by twisting
adjacent strands by full twists. The twisted torus knots lie in $F$, the genus
2 Heegaard surface for $S^3$. Primitive/primitive and primitive/Seifert knots
lie in $F$ in a particular way. Dean gives sufficient conditions for the
parameters of the twisted torus knots to ensure they are primitive/primitive or
primitive/Seifert. Using Dean's conditions, Doleshal shows that there are
infinitely many twisted torus knots that are fibered and that there are twisted
torus knots with distinct primitive/Seifert representatives with the same slope
in $F$. In this paper, we extend Doleshal's results to show there is a four
parameter family of positive twisted torus knots. Additionally, we provide new
examples of twisted torus knots with distinct representatives with the same
surface slope in $F$.
