Algebraic & Geometric Topology

The stable $4$–genus of knots

Charles Livingston

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We define the stable 4–genus of a knot KS3, 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=CQ. 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.

Article information

Algebr. Geom. Topol., Volume 10, Number 4 (2010), 2191-2202.

Received: 8 September 2010
Accepted: 12 September 2010
First available in Project Euclid: 19 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

knot concordance four-genus


Livingston, Charles. The stable $4$–genus of knots. Algebr. Geom. Topol. 10 (2010), no. 4, 2191--2202. doi:10.2140/agt.2010.10.2191.

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