Vassiliev invariants and rational knots of unknotting number one

Introducing a way to modify knots using n-trivial rational tangles, we show that knots with given values of Vassiliev invariants of bounded degree can have arbitrary unknotting number (extending a recent result of Ohyama, Taniyama and Yamada). The same result is shown for 4-genera and finite reductions of the homology group of the double branched cover. Closer consideration is given to rational knots, where it is shown that the number of n-trivial rational knots of at most k crossings is for any n asymptotically at least C(ln k)2 for any C<e2\,\ln\,2.