The Concordance Genus

For knot K, the concordance genus is the least genus of among all knots concordant to K. Casson gave the first example of a knot of concordance genus greater than the 4-ball genus, by demonstrating that the knot 6₂ has 4-ball genus 1, since it has unknotting number 1, but it cannot be concordant to a knot of genus 1, for the following reason. Its Alexander polynomial is irreducible, of degree 4. If it were concordant to a knot of genus 1, its polynomial, times a polynomial of degree at most 2, would factor as f(t)*f(t^-1), clearly an impossibility.

For other results concerning the concordance genus, see the reference below. Most of the results for 11 crossing knots were found by John McAtee, to appear soon in a preprint.

For knots that are concordant to lower genus knots, the simplest such knot is listed in a document linked to the number in the table.

Clearly the concordance genus is dependent on the category, smooth or topological, locally flat. The first known example of this occurs for knots that are topologically slice, and thus topological concordance genus = 0, but are not smoothly slice, so smooth concordance genus > 0. Since, as of yet, this distinction is extremely rare, and we concentrate on the smooth case. (The only known distinction on the chart occurs with 11n₃₄, which is topologically slice, but we do not know its smooth concordance genus.)

References

C. Livingston, The concordance genus of knots, Algebr. Geom. Topol. 4 (2004) 1-22. Arxiv:math.GT/0107141

Further information on particular knots.