We prove that fibred knots cannot be untied with t_2k-moves, for all k >= 2. More generally, we give an upper bound on the number of two strand twist operations that allow us 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 N.

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