## Algebraic & Geometric Topology

### The stable $4$–genus of knots

Charles Livingston

#### 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.

#### Article information

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

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

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

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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