A note on trace polynomial

In this paper, we mainly study the relation of two cyclically reduced words wand w' on the condition they have the same trace polynomial (i.e., tr w = trw'). By defining an equivalence relation through such operators on words as inverse, cyclically left shift, and mirror, it is straightforward to get that w∼w' implies tr w = trw'. We show by a counter example that tr w = tr w' does not imply w ∼ w'. And in two special cases, we prove that tr w = tr w' if and only if w ∼ w'.