# Maximal Thurston-Bennequin number

The Thurston-Bennequin number, usually denoted ${\displaystyle tb}$, is an invariant of nullhomologous Legendrian knots in contact manifolds, and in particular Legendrian knots in ${\displaystyle {\mathbf {R} }^{3}}$ with the standard contact structure. It is a classical result of that ${\displaystyle tb}$ is bounded above for Legendrian knots in any given topological knot type in ${\displaystyle {\mathbf {R} }^{3}}$. The maximal Thurston-Bennequin number of a smooth knot is the largest value of ${\displaystyle tb}$ among all Legendrian representatives of the knot.
Here is a quick combinatorial definition of maximal Thurston-Bennequin number. Define a rectilinear front diagram to be a knot diagram composed of only horizontal and vertical line segments, such that at any crossing, the horizontal segment lies over the vertical segment. To any rectilinear front diagram ${\displaystyle F}$, one can associate two integers: the writhe ${\displaystyle w(F)}$, defined as for any diagram by counting the number of crossings with signs (${\displaystyle +1}$ for $\displaystyle (\overcrossing)$ and ${\displaystyle -1}$ for $\displaystyle (\undercrossing)$ ), and the cusp number ${\displaystyle c(F)}$, defined to be the number of locally upper-right corners of ${\displaystyle F}$. Next define the Thurston-Bennequin number ${\displaystyle tb(F)}$ to be ${\displaystyle w(F)-c(F)}$. Finally, the maximal Thurston-Bennequin number of a knot is the maximal value of ${\displaystyle tb(F)}$ over all rectilinear front diagrams ${\displaystyle F}$ in the knot type.
For example, the rectilinear front diagram in the figure, which is a right-handed trefoil, has ${\displaystyle w=3}$, ${\displaystyle c=2}$, and ${\displaystyle tb=1}$. In fact, the maximal Thurston-Bennequin number of the right-handed trefoil is ${\displaystyle 1}$.
In the Knot Atlas, maximal Thurston-Bennequin number is given as ${\displaystyle [a][b]}$, where ${\displaystyle a}$ and ${\displaystyle b}$ are the maximal Thurston-Bennequin numbers of the knot and its mirror, respectively. The data has been imported from the KnotInfo site (see their page on the Thurston-Bennequin number).