  The m,n Turk's Head Knot, THK(m,n), is an "alternating (m,n) torus knot." We
prove the Harary-Kauffman conjecture for all THK(m,n) except for the case where
m \geq 5 is odd and n \geq 3 is relatively prime to m. We also give evidence in
support of the conjecture in that case. Our proof rests on the observation that
none of these knots have prime determinant except for THK(m,2) when P_m is a
Pell prime.
