Geodesic
The stable 4–genus of knots
Livingston, Charles1
1Department of Mathematics, Indiana University, Rawles Hall, Bloomington IN 47405-5701, USA
Algebraic and Geometric Topology, Tome 10 (2010) no. 4, pp. 2191-2202
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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.

DOI : 10.2140/agt.2010.10.2191
Keywords: knot concordance, four-genus

Livingston, Charles  1

1 Department of Mathematics, Indiana University, Rawles Hall, Bloomington IN 47405-5701, USA 
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Livingston, Charles. The stable 4–genus of knots. Algebraic and Geometric Topology, Tome 10 (2010) no. 4, pp. 2191-2202. doi: 10.2140/agt.2010.10.2191

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