Quasipositivity as an obstruction to sliceness

Source abstract

For an oriented link L ⊂ S 3 = ∂ D 4 L\,\, \subset \,\,{S^3}\, = \,\,\partial {D^4} , let χ s ( L ) {\chi _s}{\text {(}}L{\text {)}} be the greatest Euler characteristic χ ( F ) \chi (F) of an oriented 2-manifold F (without closed components) smoothly embedded in D 4 {D^4} with boundary L . A knot K is slice if χ s ( K ) = 1 {\chi _s}(K) = 1 . Realize D 4 {D^4} in C 2 {\mathbb {C}^2} as { ( z , w ) : | z | 2 + | w | 2 ≤ 1 } \{ (z,w):|z{|^2} + |w{|^2} \leq 1\} . It has been conjectured that, if V is a nonsingular complex plane curve transverse to S 3 {S^3} , then χ s ( V ∩ S 3 ) = χ ( V ∩ D 4 ) {\chi _s}(V \cap {S^3}) = \chi (V \cap {D^4}) . Kronheimer and Mrowka have proved this conjecture in the case that V ∩ D 4 V \cap {D^4} 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 P ( − 3 , 5 , 7 ) \mathcal {P}( - 3,5,7) ; all knots obtained from a positive trefoil O { 2 , 3 } O\{ 2,3\} by iterated untwisted positive doubling). As a sidelight, I give an optimal counterexample to the "topologically locally-flat Thom conjecture".