Bennequin's Conjecture

A Braid with $M$ strands and $R$ components with $P$ positive crossings and $N$ negative crossings satisfies

$$\vert P-N\vert \leq 2U+M-R\leq P+N,$$

where $U$ is the Unknotting Number. While the second part of the Inequality was already known to be true (Boileau and Weber, 1983, 1984) at the time the conjecture was proposed, the proof of the entire conjecture was completed using results of Kronheimer and Mrowka on Milnor's Conjecture (and, independently, using Menasco's Theorem).

See also Braid, Menasco's Theorem, Milnor's Conjecture, Unknotting Number

References

Bennequin, D. ``L'instanton gordien (d'après P. B. Kronheimer et T. S. Mrowka).'' Astérisque 216, 233-277, 1993.

Birman, J. S. and Menasco, W. W. ``Studying Links via Closed Braids. II. On a Theorem of Bennequin.'' Topology Appl. 40, 71-82, 1991.

Boileau, M. and Weber, C. ``Le problème de J. Milnor sur le nombre gordien des nuds algébriques.'' Enseign. Math. 30, 173-222, 1984.

Boileau, M. and Weber, C. ``Le problème de J. Milnor sur le nombre gordien des nuds algébriques.'' In Knots, Braids and Singularities (Plans-sur-Bex, 1982). Geneva, Switzerland: Monograph. Enseign. Math. Vol. 31, pp. 49-98, 1983.

Cipra, B. What's Happening in the Mathematical Sciences, Vol. 2. Providence, RI: Amer. Math. Soc., pp. 8-13, 1994.

Kronheimer, P. B. ``The Genus-Minimizing Property of Algebraic Curves.'' Bull. Amer. Math. Soc. 29, 63-69, 1993.

Kronheimer, P. B. and Mrowka, T. S. ``Gauge Theory for Embedded Surfaces. I.'' Topology 32, 773-826, 1993.

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.