In the mathematical theory of knots, the Thurston–Bennequin number, or Bennequin number, is an invariant associated with a Legendrian knot in a three dimensional contact manifold. It is named after William Thurston and Daniel Bennequin. The Thurston-Bennequin number measures the "twisting of the contact structure around the knot". Together with the rotation number, they are often referred as the "classical" invariants of Legendrian knots. The Thurston-Bennequin number of a Legendrian knot K {\displaystyle K} is usually denoted by t b ( K ) {\displaystyle \mathrm {tb} (K)} . The maximal Thurston–Bennequin number, t b ¯ ( K ) {\displaystyle {\overline {\mathrm {tb} }}(K)} , over all Legendrian representatives of a knot in R 3 {\displaystyle \mathbb {R} ^{3}} is a topological knot invariant.
Definition and properties Let K {\displaystyle K} be a null-homologous oriented Legendrian knot in a co-oriented three-dimensional contact manifold ( M 3 , ξ ) {\displaystyle (M^{3},\xi )} and fix a Seifert surface Σ {\displaystyle \Sigma } to K {\displaystyle K} , that is an embedded connected, compact, orientable surface with boundary ∂ Σ = K {\displaystyle \partial \Sigma =K} . The Thurston-Bennequin number of K {\displaystyle K} relative to Σ {\displaystyle \Sigma } is the defined as the signed intersection number of the contact plane field ξ {\displaystyle \xi } with Σ {\displaystyle \Sigma } . Let K ′ {\displaystyle K'} be a small push-off of K {\displaystyle K} obtained by pushing along a vector field v {\displaystyle v} transverse to ξ {\displaystyle \xi } . The Thurston-Bennequin number can also be defined as l k ( K , K ′ ) {\displaystyle \mathrm {lk} (K,K')} , where l k {\displaystyle \mathrm {lk} } denotes the linking number.
The Euclidean case We consider the case where ( M , ξ ) = ( R 3 , ξ s t d ) {\displaystyle (M,\xi )=(\mathbb {R} ^{3},\xi _{\mathrm {std} })} is the standard contact structure on R 3 {\displaystyle \mathbb {R} ^{3}} . If we denote ( x , y , z ) {\displaystyle (x,y,z)} the coordinates in R 3 {\displaystyle \mathbb {R} ^{3}} , the contact structure ξ s t d {\displaystyle \xi _{\mathrm {std} }} is the kernel of the one-form d z − y d x {\displaystyle dz-ydx} . The applications Π : R 3 → R 2 , ( x , y , z ) ↦ ( x , z ) {\displaystyle \Pi \colon \mathbb {R} ^{3}\to \mathbb {R} ^{2},(x,y,z)\mapsto (x,z)} and π L : R 3 → R 2 , ( x , y , z ) ↦ ( x , y ) {\displaystyle \pi _{L}\colon \mathbb {R} ^{3}\to \mathbb {R} ^{2},(x,y,z)\mapsto (x,y)} denote respectively the front projection and the Lagrangian projection. The Thurston-Bennequin number can be computed easily from its front and Lagrangian projections.
Lagrangian projection description The Thurston-Bennequin number of a Legendrian knot K ⊂ R 3 {\displaystyle K\subset \mathbb {R} ^{3}} is the writhe of its Lagrangian projection π L ( K ) {\displaystyle \pi _{L}(K)} .
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