The stable 4-genus of knots

Charles Livingston

Published 2009-04-20, updated 2010-08-12Version 4

The stable 4-genus of a knot K in 3-space is the limiting value of g_4(nK)/n, where g_4 denotes the 4-genus and n goes to infinity. This induces a seminorm on CQ, the concordance group tensored with the rational numbers. Basic properties of the stable genus are developed, as are examples focused on understanding the unit ball under this seminorm for 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 the stable genus can be a noninteger, something that cannot be detected by classical invariants or those arising from Heegaard-Floer or Khovanov homology. It is unknown if the stable genus determines a norm on CQ and no noninteger value for the stable genus is known.