In the mathematical theory of knots, the Thurston–Bennequin number, or Bennequin number, of a front diagram of a

Legendrian knot In mathematics, a Legendrian knot often refers to a smooth embedding of the circle into which is tangent to the standard contact structure In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hy ...

is defined as the

writhe In knot theory, there are several competing notions of the quantity writhe, or \operatorname. In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amou ...

of the diagram minus the number of right

cusp A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth. Cusp or CUSP may also refer to: Mathematics * Cusp (singularity), a singular point of a curve * Cusp catastrophe, a branch of bifurca ...

s. It is named after

William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston ...

and Daniel Bennequin. The maximum Thurston–Bennequin number over all Legendrian representatives of a knot is a topological

knot invariant In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some i ...

References

* * {{knottheory-stub Knot invariants