Embedding tangles in links and Gaussian integers

We give a necessary condition for a tangle to be embedded in a link by using a greatest common devisor of Gaussian integers obtained from the bracket polynomials of links induced from the tangle. For each link L, we can associate a polynomial [Formula: see text] with integer coefficients arising from the Kauffman bracket polynomial [Formula: see text] of L. Let T be an [Formula: see text]-tangle and [Formula: see text] is a basis for the Temperley–Lieb algebra of dimension n. For a Gaussian integer [Formula: see text], let d be a greatest common devisor of [Formula: see text], where [Formula: see text] is the closure of the product of the two tangles T and [Formula: see text] for [Formula: see text]. If the tangle T is embedded in a link L, then we show that d is a divisor of [Formula: see text] for some natural number k.