Rotation number (knot theory)

In the mathematical theory of knots, the rotation number is an invariant associated with knots that are adapted to a contact structure, such as Legendrian and transverse knots, in a three-dimensional contact manifold. It measures the twisting of the knot's tangent direction relative to the contact structure. Together for the Thurston–Bennequin number for Legendrian knots, it is often referred to as a "classical" invariant of Legendrian knots.

The rotation number of a knot ${\displaystyle K}$ is commonly denoted by ${\displaystyle \operatorname {rot} (K)}$ or ${\displaystyle r(K)}$.