We examine the Kauffman bracket expansion of the generalized crossing Δn, a half-twist on n parallel strands, as an element of the Temperley-Lieb algebra with coeffcients in Z[A;A^-1]. In particular, we determine the minimum and maximum degrees of all possible coeffcients appearing in this expansion. Our main theorem shows that the maximum such degree is quadratic in n, while the minimum such degree is linear. We also include an appendix with explicit expansions for n at most six
