Women who have a family history of fraternal twins have a higher chance of producing fraternal twins themselves, as there is a genetically linked tendency to hyper-ovulate.
There is no known genetic link for identical twinning.
Epigenetics refers to the level of activity of any particular gene.
Monozygotic twins, although genetically very similar, are not genetically exactly the same.
The children of monozygotic twins test genetically as half-siblings (or full siblings, if a pair of monozygotic twins reproduces with another pair or with the same person), rather than first cousins.
Identical twins do not have the same fingerprints however, because even within the confines of the womb, the fetuses touch different parts of their environment, giving rise to small variations in their corresponding prints and thus making them unique.
Dizygotic twinning ranges from six per thousand births in Japan (similar to the rate of monozygotic twins) to 14 and more per thousand in some African countries.
Comparison of zygote development in monozygotic and dizygotic twins.