Slice genus

In mathematics, the slice genus of a smooth knot "K" in "S³" (sometimes called its "Murasugi genus" or "4-ball genus") is the least integer g such that "K" is the boundary of a connected, orientable 2-manifold "S" of genus "g" embedded in the 4-ball "D⁴" bounded by "S³".

More precisely, if "S" is required to be smoothly embedded, then this integer "g" is the "smooth slice genus" of "K" and is often denoted g_s("K") or g₄("K"), whereas if "S" is required only to be topologically locally flatly embedded then "g" is the "topologically locally flat slice genus" of "K". (There is no point considering "g" if "S" is required only to be a topological embedding, since the cone on "K" is a 2-disk with genus 0.) There can be an arbitrarily great difference between the smooth and the topologically locally flat slice genus of a knot; a theorem of Michael Freedman says that if the Alexander polynomial of "K" is 1, then the topologically locally flat slice genus of "K" is 0, but it can be proved in many ways (originally with gauge theory) that for every g there exist knots "K" such that the Alexander polynomial of "K" is 1 while the genus and the smooth slice genus of "K" both equal g.

The (smooth) slice genus of a knot "K" is bounded below by a quantity involving the Thurston-Bennequin invariant of "K":

: $g\_s(K)\; ge\; (\{\; m\; TB\}(K)+1)/2\; ,\; .$

The (smooth) slice genus is zero if and only if the knot is concordant to the unknot.

References

*cite journal
author = Rudolph, Lee
title = The slice genus and the Thurston-Bennequin invariant of a knot
journal = Proceedings of the American Mathematical Society
volume = 125
pages = 3049 3050
year = 1997
id = MathSciNet | id = 1443854
doi = 10.1090/S0002-9939-97-04258-5

* Livingston, Charles, A survey of classical knot concordance, in: "Handbook of knot theory", pp 319–347, Elsevier, Amsterdam, 2005. MathSciNet | id = 2179265 ISBN 0-444-51452-X