Thurston–Bennequin number

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Let ${\displaystyle K}$ be a null-homologous oriented Legendrian knot in a co-oriented three-dimensional contact manifold ${\displaystyle (M^{3},\xi )}$ and fix a Seifert surface ${\displaystyle \Sigma }$ to ${\displaystyle K}$, that is an embedded connected, compact, orientable surface with boundary ${\displaystyle \partial \Sigma =K}$. The Thurston-Bennequin number of ${\displaystyle K}$ relative to ${\displaystyle \Sigma }$ is the defined as the signed intersection number of the contact plane field ${\displaystyle \xi }$ with ${\displaystyle \Sigma }$.^[3]

Let ${\displaystyle K'}$ be a small push-off of ${\displaystyle K}$ obtained by pushing along a vector field ${\displaystyle v}$ transverse to ${\displaystyle \xi }$. The Thurston-Bennequin number can also be defined as ${\displaystyle \mathrm {lk} (K,K')}$, where ${\displaystyle \mathrm {lk} }$ denotes the linking number.^[3]

The Euclidean case

We consider the case where ${\displaystyle (M,\xi )=(\mathbb {R} ^{3},\xi _{\mathrm {std} })}$ is the standard contact structure on ${\displaystyle \mathbb {R} ^{3}}$. If we denote ${\displaystyle (x,y,z)}$ the coordinates in ${\displaystyle \mathbb {R} ^{3}}$, the contact structure ${\displaystyle \xi _{\mathrm {std} }}$ is the kernel of the one-form ${\displaystyle dz-ydx}$. The applications ${\displaystyle \Pi \colon \mathbb {R} ^{3}\to \mathbb {R} ^{2},(x,y,z)\mapsto (x,z)}$ and ${\displaystyle \pi _{L}\colon \mathbb {R} ^{3}\to \mathbb {R} ^{2},(x,y,z)\mapsto (x,y)}$ 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 ${\displaystyle K\subset \mathbb {R} ^{3}}$ is the writhe of its Lagrangian projection ${\displaystyle \pi _{L}(K)}$.

Front projection description

For a Legendrian knot ${\displaystyle K\subset \mathbb {R} ^{3}}$, its front projection ${\displaystyle \Pi (K)\subset \mathbb {R} ^{2}}$ is called its front diagram. The front diagram of a Legendrian knot does not have vertical tangencies, however cusps can appear. Generically, the front diagram of a knot as no tangency point, no triple intersection and standard cusp singularities. In this case the Thurston-Bennequin number is

${\displaystyle \mathrm {tb} (K)={\textrm {writhe}}(\Pi (K))-{\dfrac {1}{2}}{\big (}\#{\text{number of cusps}}{\big )},}$

where ${\displaystyle {\textrm {writhe}}(\Pi (K))}$ denotes the writhe of the front diagram.^[1]

The invariant can also be computed using a grid diagram corresponding to a particular Legendrian representative of a knot.^[4][5] 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.

In his thesis ^[1], Daniel Bennequin proved an inequality involving the Thurston-Bennequin number. He proved that for all Legendrian knot ${\displaystyle K}$ in the standard contact ${\displaystyle \mathbb {R} ^{3}}$ the following inequality is true:

${\displaystyle \mathrm {tb} (K)+\vert {\textrm {rot}}(K)\vert \leq -\chi (\Sigma ),}$

where ${\displaystyle \chi (\Sigma )}$ denotes the Euler characteristic of a Seifert surface ${\displaystyle \Sigma }$ of ${\displaystyle K}$ and ${\displaystyle {\textrm {rot}}(K)}$ denotes the rotation number of ${\displaystyle K}$.

In particular, the maximal Thurston-Bennequin number gives a lower bound on the genus of a topological knot.