Two strand twisting

We prove that fibred knots cannot be untied with ${\bar{t}}_{2k}$-moves, for all $k\ge 2$. More generally, we give an upper bound on the number of two strand twist operations that allow to untie a knot with non-trivial HOMFLY polynomial, in terms of the minimal crossing number, and the braid index. As a by-product, we prove that the braid index of a two-bridge knot cannot be lowered by applying $t_{2k}$-moves, for all but finitely many $k\in \mathbb{N}$.