Thurston–Bennequin number

In the mathematical theory of knots, the Thurston–Bennequin number, or Bennequin number, of a front diagram of a Legendrian knot is defined as the writhe of the diagram minus the number of right cusps^[1]. It is named after William Thurston and Daniel Bennequin.

The Thurston-Bennequin number of a knot ${\displaystyle K}$ is commonly denoted by ${\displaystyle \mathrm {tb} (K)}$. The maximal Thurston–Bennequin number, ${\displaystyle {\overline {\mathrm {tb} }}(K)}$, over all Legendrian representatives of a knot is a topological knot invariant^[2].

The invariant can also be computed using a grid diagram corresponding to a particular Legendrian representative of a knot^[3][4]. In this setting, the number can be computed as the writhe of the diagram minus the number of 'northwest' corners.

A grid diagram of the knot ${\displaystyle 8_{20}}$ and an associated Legendrian representative of it.

By smoothing the 'northeast' and 'southwest' corners and rotating the diagram and switching all crossings, one can convert a grid diagram into the associated Legendrian knot.