Thurston–Bennequin number

Let K be a null-homologous oriented Legendrian knot in a co-oriented three-dimensional contact manifold (M^3,\xi) and fix a Seifert surface \Sigma to K, that is an embedded connected, compact, orientable surface with boundary \partial \Sigma = K. The Thurston-Bennequin number of K relative to \Sigma is the defined as the signed intersection number of the contact plane field \xi with \Sigma. The invariant can also be computed using a grid diagram corresponding to a particular Legendrian representative of a knot. In this setting, the number can be computed as the writhe of the diagram minus the number of 'northwest' corners. 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. ==The Bennequin inequality==

In his thesis , Daniel Bennequin proved an inequality involving the Thurston-Bennequin number. He proved that for all Legendrian knot K in the standard contact \mathbb R^3 the following inequality is true: : \mathrm{tb}(K)+\vert \textrm{rot}(K) \vert \leq - \chi(\Sigma), where \chi(\Sigma) denotes the Euler characteristic of a Seifert surface \Sigma of K and \textrm{rot}(K) denotes the rotation number of K . In particular, the maximal Thurston-Bennequin number gives a lower bound on the genus of a topological knot. ==References==