Algebraic & Geometric Topology

The stable $4$–genus of knots

Charles Livingston

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715167

Digital Object Identifier
doi:10.2140/agt.2010.10.2191

Mathematical Reviews number (MathSciNet)
MR2745668

Zentralblatt MATH identifier
1213.57015

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

Keywords
knot concordance four-genus

Citation

Livingston, Charles. The stable $4$–genus of knots. Algebr. Geom. Topol. 10 (2010), no. 4, 2191--2202. doi:10.2140/agt.2010.10.2191. https://projecteuclid.org/euclid.agt/1513715167


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