TITLE: Primitive elements in a free group of rank three 
QUESTION [10 upvotes]: It is well-known that the fundamental group of a twice-punctured torus is a free group of rank three. 
I see that there is no one-to-one correspondence between the homotopy classes of essential simple loops on twice-punctured torus and the conjugacy classes of primitive elements in a free group of rank three. 
Do we know which primitive elements in a free group of rank three represent simple loops on a twice-punctured torus?

REPLY [6 votes]: There are 3 types of simple closed curves on a twice punctured torus.
The first type are two isotopy classes of curves which are isotopic to the two
boundary components. These are primitive and separating. 
The second type is separating curves which cut the torus into
a once-punctured torus and a pair of pants. These curves are not
primitive, since they are commutators (although I assume this
is not what you mean when you observe there is no 1-1 correspondence
between primitive curves and simple closed curves).
The third type is non-separating curves which are primitive. These
curves have the property that they are sent to their inverse
(when keeping track of orientation) under the elliptic involution (which
exchanges the two boundary components).
So this property is certainly a restriction: a primitive curve conjugate
to a simple closed curve
which is not isotopic to a boundary component must be sent to
its inverse under the elliptic involution. I think this symmetry 
condition is not sufficient, since the subgroup of $Aut(F_3)$
commuting with this involution is larger than the mapping torus of 
the twice-punctured torus.