New topologically slice knots

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 $ℤ ⋉ ℤ [1 ∕ 2]$ . 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).

Keywords

slice knots, surgery, Blanchfield pairing

Mathematical Subject Classification 2000

Primary: 57M25

Secondary: 57M27, 57N70

References

Forward citations

Publication

Received: 12 May 2005

Accepted: 10 October 2005

Published: 4 November 2005

Proposed: Robion Kirby

Seconded: Cameron Gordon, Wolfgang Lueck

Correction: 18 October 2006

Authors

Stefan Friedl
Department of Mathematics Rice University Houston Texas 77005 USA

Peter Teichner
Department of Mathematics University of California Berkeley California 94720 USA

Volume 9, issue 4 (2005)

Stefan Friedl and Peter Teichner

Abstract