An obstruction to embedding 4-tangles in links

We consider the ways in which a 4-tangle T inside a. unit cube can be exten ded outside the cube into a knot or link L. We present two links n(T) and d (T) such that the greatest common divisor of the determinants of these two links always divides the determinant of the link L. In order to prove this result we give a two-integer invariant of 4-tangles. Calculations are facilitated by viewing the determinant as the Kauffman br acket at a fourth root of -1, which sets the loop factor to zero. For ratio nal tangles, our invariant coincides with the value of the associated conti nued fraction.