Thurston-Bennequin number
Encyclopedia
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 cusp
s. It is named after William Thurston
and Daniel Bennequin.
The maximum Thurston–Bennequin number over all Legendrian representatives of a knot is a topological knot invariant
.
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, the Thurston–Bennequin number, or Bennequin number, of a front diagram of a Legendrian knot
Legendrian knot
In mathematics, a Legendrian knot often refers to a smooth embedding of the circle into \mathbb R^3, which is tangent to the standard contact structure on \mathbb R^3...
is defined as the writhe
Writhe
In knot theory, the writhe is a property of an oriented link diagram. The writhe is the total number of positive crossings minus the total number of negative crossings....
of the diagram minus the number of right cusp
Cusp (singularity)
In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
s. It is named after William Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
and Daniel Bennequin.
The maximum Thurston–Bennequin number over all Legendrian representatives of a knot is a topological knot invariant
Knot invariant
In the mathematical field of knot theory, a knot invariant is a quantity 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 invariants are indeed numbers, but invariants can range from the...
.