Quasipositivity as an obstruction to sliceness

Abstract

Abstract. For an oriented link L C S3 = dD4 , let Xs(L) be the greatest Euler characteristic x(F) of an oriented 2-manifold F (without closed components) smoothly embedded in D4 with boundary L . A knot K is slice if Xs(K) = 1 . Realize D4 in C2 as {(z, w) : \z\2 + \w\2 < 1} . It has been conjectured that, if V is a nonsingular complex plane curve transverse to S3 , then Xs(VnS3) = ^(KnZ)4). Kronheimer and Mrowka have proved this conjecture in the case that V n D4 is the Milnor fiber of a singularity. I explain how this seemingly special case implies both the general case and the "slice-Bennequin inequality" for braids. As applications, I show that various knots are not slice (e.g., pretzel knots like ^(-3, 5,7); all knots obtained from a positive trefoil 0{2, 3} by iterated untwisted positive doubling). As a sidelight, I give an optimal counterexample to the "topologically locally-flat Thorn conjecture". A BRIEF HISTORY OF SLICENESSA link is a compact 1-manifold without boundary L (i.e., finite union of simple closed curves) smoothly embedded in the 3-sphere S3 ; a knot is a link with one component. If S3 is realized in R4 as, say, the unit sphere, then a natural way to construct links is to intersect suitable two-dimensional subsets X c R4 with S3 ; one may then ask how constraints on X are reflected in constraints on the link X r\S3.For instance, Fox and Milnor (c. 1960) considered, in effect, the case that X is a smooth 2-sphere intersecting S3 transversally; at Moise's suggestion, Fox [5] adopted the adjective slice to describe the knots and links X nS3 so constructed. Fox and Milnor [6] gave a criterion for a knot A" to be slice: its Alexander polynomial Ak(í) £ Z[t, t~x] must have the form F(t)F(rx). This shows that, for instance, the two trefoil knots 0{2, ±3} are not slice (since A0{2,±3} = t~x -l+t is not of the form F(t)F(t~x) ), but it says nothing about the two granny knots 0{2, 3}#0{2, 3}, 0{2, -3}#0{2, -3} (indeed, both granny knots share the Alexander polynomial (r-1 -1 + i)2 with the square

Embedded surfaces in four-manifolds, branched covers, and SO(3)-invariants

“…Lee and Wilczynski [17] have shown that the lower bounds proved in [8], [20] are realized by topologically locally flat surfaces, for all d which are even or powers of an odd prime. There were earlier counterexamples to the topologically locally flat version of the Thom conjecture, due to Rudolph [21]; see also [22]. It follows from Theorem 1-1 that some of these are not smoothable, cf.…”

“…By Theorem 1-1, this is not smoothable for d = 6. In [22], there is an example of a topologically locally flat embedding of a surface of genus 5 representing 5 times the generator of H 2 (CP 2 ,Z), which has geometric intersection number 5 with a complex line. This, too, cannot be smoothable, since by replacing the intersection points with handles, one would obtain an embedded surface of degree 6 and genus 9, contradicting Theorem 1-1.…”

A criterion for the Legendrian simplicity of the connected sum

“…This result has been improved in many ways, related to classical link invariants such as genus [1,15], polynomials [11,14], or other invariants such as Khovanov homology [12], knot Floer homology [13] and so on. However they are not essential in this paper and we omit the detail.…”

Relations among the lowest degree of the Jones polynomial and geometric invariants for a closed positive braid

“…We shall call it the Bennequin unknotting inequality. In [15], by using a result of Kronheimer and Mrowka [9], [10], Rudolph showed the slice-Bennequin inequality,…”