Thurston–Bennequin number

Short description: Mathematical theory of knots

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 $K$ is usually denoted by $\mathrm{t}\mathrm{b}(K)$. The maximal Thurston–Bennequin number, $\overline{\mathrm{t}\mathrm{b}}(K)$, over all Legendrian representatives of a knot in ${\mathbb{ℝ}}^3$ is a topological knot invariant.

Definition and properties

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

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

The Euclidean case

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

Front projection description

For a Legendrian knot $K\subset {\mathbb{ℝ}}^3$, its front projection $\mathrm{\Pi }(K)\subset {\mathbb{ℝ}}^2$ 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

$\mathrm{t}\mathrm{b}(K)=\text{<mrow data-mjx-texclass="ORD">writhe</mrow>}(\mathrm{\Pi }(K))-\frac{1}{2}\left(\mathrm{\#}\text{number of cusps}\right),$

where $\text{<mrow data-mjx-texclass="ORD">writhe</mrow>}(\mathrm{\Pi }(K))$ denotes the writhe of the front diagram.

The invariant can also be computed using a grid diagram corresponding to a particular Legendrian representative of a knot. In this setting, the number can be computed as the writhe of the diagram minus the number of 'northwest' corners.

A grid diagram of the knot $8_{20}$ 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.

The Bennequin inequality

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

$\mathrm{t}\mathrm{b}(K)+|\text{<mrow data-mjx-texclass="ORD">rot</mrow>}(K)|\le -\chi (\mathrm{\Sigma }),$

where $\chi (\mathrm{\Sigma })$ denotes the Euler characteristic of a Seifert surface $\mathrm{\Sigma }$ of $K$ and $\text{<mrow data-mjx-texclass="ORD">rot</mrow>}(K)$ denotes the rotation number of $K$.

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

References

• "Thurston–Bennequin number", The Knot Atlas.
• Lee Rudolph (1997). "The slice genus and the Thurston–Bennequin invariant of a knot". Proceedings of the American Mathematical Society 125: 3049–3050. doi:10.1090/S0002-9939-97-04258-5.