A remarkable 20-crossing tangle

For any positive integer r, we exhibit a knot Kr with (20 $\times$ 2 r–1 +

1. crossings whose Jones polynomial V (Kr) is equal to 1 mod-ulo 2 r. Our construction rests on a certain 20-crossing tangle T 20 which is undetectable by the Kauffman bracket polynomial pair mod 2.