Vassiliev invariants and rational knots of unknotting number one

Abstract

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 $C<\sqrt[2\ln 2]{e}$.