New topologically slice knots

Details New topologically slice knots New topologically slice knots

2005-05-12

arxiv.org/abs/math/0505233v3

Geom. Topol. 9(2005) 2129-2158 and Geom. Topol. 10(2006) 2501--2504

Mathematics

Geometric Topology

This is the version published by GT on 4 November 2005 and includes the erratum published on 18 October 2006

Scientific paper

10.2140/gt.2005.9.2129 10.2140/g

In the early 1980's Mike Freedman showed that all knots with trivial Alexander polynomial are topologically slice (with fundamental group Z). This paper contains the first new examples of topologically slice knots. In fact, we give a sufficient homological condition under which a knot is slice with fundamental group Z semi-direct product Z[1/2]. These two fundamental groups are known to be the only solvable ribbon groups. Our homological condition implies that the Alexander polynomial equals (t-2)(t^{-1}-2) but also contains information about the metabelian cover of the knot complement (since there are many non-slice knots with this Alexander polynomial). Erratum (attached): In Figure 1.5 we gave an incorrect example for Theorem 1.3. We present a correct example.

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