Thurston–Bennequin number

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". ^[1] Together with the rotation number, they are often referred as the "classical" invariants of Legendrian knots.

The Thurston-Bennequin number of a Legendrian knot ${\displaystyle K}$ is usually denoted by ${\displaystyle \mathrm {tb} (K)}$. The maximal Thurston–Bennequin number, ${\displaystyle {\overline {\mathrm {tb} }}(K)}$, over all Legendrian representatives of a knot in ${\displaystyle \mathbb {R} ^{3}}$ is a topological knot invariant. ^[2]

Definition and properties

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.

A grid diagram of the knot ${\displaystyle 8_{20}}$ 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 ^[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.

References

1. "Entrelacements et équations de Pfaff". Astérisque. 107/108: 87–161. 1983. (Bennequin's doctoral dissertation)
2. Ng, Lenhard (2012). "On arc index and maximal thurston–bennequin number". Journal of Knot Theory and Its Ramifications. 21 (04): 1250031. arXiv: math/0612356 . doi:10.1142/S0218216511009820. ISSN   0218-2165.
3. Geiges, Hansjörg (2008). An introduction to contact topology; Volume 109 of Cambridge studies in advanced mathematics. Cambridge University Press. p. 94. ISBN   978-0-521-86585-2.
4. Ozsváth, Peter S.; Stipsicz, András I.; Szabó, Zoltán (2015). Grid Homology for Knots and Links. American Mathematical Society. pp. 220–221. ISBN   978-1-4704-3442-7.
5. Dynnikov, I.; Prasolov, M. (2013). "Bypasses for rectangular diagrams. A proof of the Jones conjecture and related questions". Transactions of the Moscow Mathematical Society. 74: 97–144. arXiv: 1206.0898 . doi:10.1090/S0077-1554-2014-00210-7. ISSN   0077-1554.

• " 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 . MR   1443854.