Menasco's Theorem

For a Braid with $M$ strands, $R$ components, $P$ positive crossings, and $N$ negative crossings,

$$\cases{ P-N \leq U_++M-R & if $P\geq N$\cr P-N \leq U_-+M-R & if $P\leq N$,\cr}$$

where $U_\pm$ 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. Menasco's theorem can be extended to arbitrary knot diagrams.

See also Bennequin's Conjecture, Braid, Unknotting Number

References

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

Menasco, W. W. ``The Bennequin-Milnor Unknotting Conjectures.'' C. R. Acad. Sci. Paris Sér. I Math. 318, 831-836, 1994.