Volume 10, issue 4 (2010)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
The stable $4$–genus of knots

Charles Livingston

Algebraic & Geometric Topology 10 (2010) 2191–2202
Abstract

We define the stable 4–genus of a knot K S3, gst(K), to be the limiting value of g4(nK)n, where g4 denotes the 4–genus and n goes to infinity. This induces a seminorm on the rationalized knot concordance group, CQ = C Q. Basic properties of gst are developed, as are examples focused on understanding the unit ball for gst on specified subspaces of CQ. 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 gst(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