2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/79080Introducing 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 $C<\sqrt[2\ln 2]{e}$.13 pages, 2 figures; revision 26 Nov 99: added reference [OTY], discussion of signatures, branched cover homology and 4-genera, more problems; revision 7 Sep 01: Theorem 1.2 slightly improved, a few other minor structural changes; revision 20 Dec 01: final version, Theorem 1.2 improved, 2 sections removedGeometric Topology57M25text