Vassiliev invariants and rational knots of unknotting number one

Details Vassiliev invariants and rational knots of unknotting number one Vassiliev invariants and rational knots of unknotting number one

1999-09-09

arxiv.org/abs/math/9909050v4

Topology 42(1) (2003), 227--241.

Mathematics

Geometric Topology

13 pages, 2 figures; revision 26 Nov 99: added reference [OTY], discussion of signatures, branched cover homology and 4-genera

Scientific paper

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

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