Abstract: Let Mbe a 3-manifold obtained by surgery on then-component framed link LS^{3. We can get the link invariant maintaining isotopy, and K₊ and K_- moves for the framed link (L, f),so as to furthermoreobtain the invariant for 3-manifold. Thus the invariants for 3-manifold are given by finite linear combinations of Kauffman brackets of cabling of L, to be evaluated at primitive 4r-th root of unity. Some difference representations of invariant for 3-manifold are given by using the bracket polynomial, TemperlyLieb algebra, JonesKauffman module and linear skein theory, etc. This paper deals with the relation of these representations of invariants, and shows essentially the consistency of them. A new representation of the invariant of 3-manifolds is also given.}

Key words: framed link, invariant, bracket polynomial, Jones-Kauffman module