For a braid with strands, components, positive crossings, and negative crossings,
(1)
where
are the smallest number of positive and negative crossings which must be changed
to crossings of the opposite sign. These inequalities imply Bennequin's
conjecture. This inequality can also be extended to arbitrary knot diagrams.
Menasco (1994) published a purported purely three-dimensional proof of the theorem that was discussed in Cipra (1994) and Menasco and Rudolph (1995). However, a hole in the proof was subsequently discovered by Otal. The hole has not yet been patched up, so the only proof of the inequality is the one due to Rudolph (1993), building on work by Kronheimer and Mrowka in four-dimensional topology.
Cipra, B. "From Knot to Unknot." What's Happening in the Mathematical Sciences, Vol. 2. Providence, RI: Amer.
Math. Soc., pp. 8-13, 1994.Kronheimer, P. B. and Mrowka, T. S.
"Recurrence Relations and Asymptotics for Four-Manifold Invariants." Bull.
Amer. Math. Soc.30, 215-221, 1994.Menasco, W. W. "The
Bennequin-Milnor Unknotting Conjectures." C. R. Acad. Sci. Paris Sér.
I Math.318, 831-836, 1994.Menasco, W. W. and Rudolph,
L. "How Hard Is It to Untie a Knot?" Amer. Sci.83, 38-49,
1995.Rudolph, L. "Quasipositivity as an Obstruction to Sliceness."
Bull. Amer. Math. Soc.29, 51-59, 1993.
strands, 
components, 
positive crossings, and 
negative crossings,
are the smallest number of positive and negative crossings which must be changed
to crossings of the opposite sign. These inequalities imply Bennequin's
conjecture. This inequality can also be extended to arbitrary knot diagrams.
