The author uses the notation \(\mathbb{T}_n\) for the set of rational \(n\)-string tangles and to each diagram \(D\) of a tangle \(T\in \mathbb{T}\) associates the matrix \(M(D)(a,a^{-1})\in M_{3\times 3}(\mathbb{Z}[a,a^{-1}])\) which is a regular isotopy invariant obtained by applying the Kauffman bracket polynomial to tangles, from which the matrices \(M_1\) and \(M_2\), evaluated at \(a = \sqrt i\) and \(a=\frac{1+\sqrt 3}{2}\), respectively, are obtained.
The author embeds a rational 2-string tangle \(B\) in \(\mathbb{T}_2\) to obtain the 3-tangles \(B^g\) and \(B_g\) in \(\mathbb{T}_3\) by adding the bottom and top strings, respectively, and writes \(\mathbb{T}\mathbb{G} = \{B^g|B\in \mathbb{T}_2\}\subset \{B_g\mid B\in \mathbb{T}_2\}\). Also the author defines a subclass \(B\widehat X B\) of \(\mathbb{T}_3\) by using 4 special types of 3-tangles and the 3-braid group \(\mathbb{B}_3\).
Then the author extends the previous result on the classification of the 3-braid group \(\mathbb{B}_3\) to the subfamilies \(\mathbb{B}_3\subset \mathbb{T}\mathbb{G}\), \(\mathbb{B}_3+\mathbb{T}\mathbb{G}\), \(\mathbb{T}\mathbb{G}+\mathbb{B}_3\) and \(B\widehat XB\) of \(\mathbb{T}^3\), by using the matrix \(M_2\) and a certain equivalence class of \(M_1\) which are ambient isotopy invariants.
Theorem 0.1. Let \(T_1,T_2\in\mathbb{T}\). Then \(T_1\) and \(T_2\) are equivalent if and only if \(M_j(T_1) = M_j(T_2)\) for \(j = 1,2\) where \(\mathbb{T}\) denotes one of the subfamilies \(\mathbb{B}_3\), \(\mathbb{B}_3\cup \mathbb{T}\mathbb{G}\), \(\mathbb{B}_3+\mathbb{T}\mathbb{G}\), \(\mathbb{T}\mathbb{G} +\mathbb{B}_3\) and \(B\widehat XB\) of \(\mathbb{T}_3\).
Finally the author says that it is still not known whether these invariants will classify the set \(\mathbb{T}_3\).