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? 



Primitive elements in a free group of rank three 
A classification of primitive elements in a free group of rank greater than two is a hard problem, and there is no really satisfactory classification known. I am pretty sure [this paper of Shpilrain](http://www.sci.ccny.cuny.edu/~shpil/countprim.ps) is pretty close to the last word. As for elements representing simple closed curves, this is also not easy, and the best results are algorithmic results of D. Chillingworth (MR0248819 (40 #2069) 
Chillingworth, D. R. J.
Simple closed curves on surfaces. 
Bull. London Math. Soc. 1 1969 310–314), which were essentially replicated by Birman-Series (MR0744104 (85m:57002) 
Birman, Joan S.(1-CLMB); Series, Caroline(4-WARW)
An algorithm for simple curves on surfaces. 
J. London Math. Soc. (2) 29 (1984), no. 2, 331–342. ) and Cohen-Lustig (MR0895629 (88m:57016) 
Cohen, Marshall(1-CRNL); Lustig, Martin(1-MIT)
Paths of geodesics and geometric intersection numbers. I. Combinatorial group theory and topology (Alta, Utah, 1984), 479–500, 
Ann. of Math. Stud., 111, Princeton Univ. Press, Princeton, NJ, 1987. 
57N05 (11F06) )


**EDIT**
One relevant remark: The number of conjugacy classes of simple closed curves grows polynomially (the rates of growth were determined by I. Rivin in "Simple curves on surfaces", and corresponding asymptotic results were obtained by M. Mirzakhani. The number of conjugacy classes of primitive elements, however, grows exponentially (easy construction for $F\_3:$ take $x\_3$ times any word in $x\_1, x\_2$). This is why things become much harder past $F\_2.$


