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Thurston–Bennequin number

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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 is commonly denoted by . The maximal Thurston–Bennequin number, , 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.

Thumb
A grid diagram of the knot 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.

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