The stable 4–genus of knots

Livingston, Charles

doi:10.2140/agt.2010.10.2191

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The stable $4$–genus of knots

Charles Livingston

Algebraic & Geometric Topology 10 (2010) 2191–2202

DOI: 10.2140/agt.2010.10.2191

Abstract

We define the stable $4$ –genus of a knot $K \subset S^{3}$ , $g_{st} (K)$ , to be the limiting value of $g_{4} (n K) ∕ n$ , where $g_{4}$ denotes the $4$ –genus and $n$ goes to infinity. This induces a seminorm on the rationalized knot concordance group, $C_{Q} = C \otimes Q$ . Basic properties of $g_{st}$ are developed, as are examples focused on understanding the unit ball for $g_{st}$ on specified subspaces of $C_{Q}$ . Subspaces spanned by torus knots are used to illustrate the distinction between the smooth and topological categories. A final example is given in which Casson–Gordon invariants are used to demonstrate that $g_{st} (K)$ can be a noninteger.

Keywords

knot concordance, four-genus

Mathematical Subject Classification 2000

Primary: 57M25

References

Publication

Received: 8 September 2010

Accepted: 12 September 2010

Published: 30 October 2010

Authors

Charles Livingston
Department of Mathematics Indiana University Rawles Hall Bloomington IN 47405-5701 USA

Volume 10, issue 4 (2010)