
Undergraduate Research at GANG
Summer 2000

Curves and Immersions
The Flower Sequence 1
The sequence takes a curve with a single inner
loop and adds successive outer loops that (inner) loop. The curves in
the top row bound nothing. The curves on the bottom bound successively
higher genus manifolds. That is, the first bounds a punctured sphere
(a disk), the second a punctured torus the third a punctured
twotorus. The sequence continues in this manner. If one considers the
curves on the bottom with 1 added to each degree number, not only do
these curves bound
ntori but also (n1)tori
in multiple ways. The
number of immersions
(a_{n})
increases as n increases according to the
sequence
a_{n} = 4 a_{n1}1.

