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".