In this paper, we mainly study the relation of two cyclically reduced words w and w' on the condition they have the same trace polynomial (i.e., tr w= tr w' ). 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 = tr w'. 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'.
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