We give a necessary condition for a 4-tangle T to be embedded in a link by using a common divisor of two integers obtained from the Kauffman bracket polynomials of the numerator N(T) and the denominator D(T) of the tangle T. The Kauffman bracket polynomial of a link L is a regular isotopy invariant of links taking values in the Laurent polynomial ring ZA, A(-1). For each integer i, we associate an integer B-L (i), which is obtained from the Kauffman bracket polynomial of L. Let d be the greatest common divisor of B-N(T)(i) and B-D(T) (i) for a 4-tangle T. We will show that there exists a natural number n such that d divides i(n)B(L)(i) if the tangle T is embedded in a link L.
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