Vassiliev invariants and rational knots of unknotting number one

1999/09/30 by A. Stoimenow · 1 citation
Mathematics · #math.GT #msc:57M25

paper · pdf

published as Topology 42(1) (2003), 227--241. · 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 removed

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<√[2ln 2]e.

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• Algebraic structures of Vassiliev invariants for knot families

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