Menasco's theorem

For a braid with M strands, R components, P positive crossings, and N negative crossings, {P - N<=U_+ + M - R | if P>=N P - N<=U_- + M - R | if P<=N, auto right match where U_± 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.

Bennequin's conjecture | braid | unknotting number