I am looking for parametrizations of smooth 2-tori in $R^3$ with multiple holes (genus > 1).
For instance, Wikipedia has pictures of smooth-looking 2-holed and 3-holed tori. I was wondering what parametrizations they correspond to.
Would there be some way to generalize the process to a genus n surface?
Topologically, I understand an increase in genus by 1 is like attaching a handle. So one could take an ordinary genus-1 torus, add handles to it and somehow smooth it out locally (perhaps using bump functions ?), but it seems hard to start from the usual parametrization of the torus and actually do so for any given $n$.
(Of course I understand there are many such possibilities, I am just looking for something systematic). Thanks.