Trace polynomials of words in the free group of rank two

Abstract

Procesi's theorem guarantees that traces in a two generator subgroup of $\ssl$ are polynomials in traces of the generators. These polynomials are called trace polynomials and defined for words in the free group of rank two. Let $\cw$ denote the set of cyclically reduced words in $F_2$. Improving Jorgensen's algorithm, we classify all words in $\cw$ with the word lengths less than nine via their trace polynomials. Then we check whether they are in $\sim$-equivalence defined from the operation Mirror, Left shift, and Inverse on $\cw$. We prove that two words of the same trace polynomials are $\sim$-equivalent when the word lengths are less than nine. We also show, by counterexamples, this result does not hold for the word lengths greater than eight. As a corollary, we verify Wang's conjecture for the word lengths less than nine.