# Maximal Thurston-Bennequin number

The Thurston-Bennequin number, usually denoted , is an invariant of nullhomologous Legendrian knots in contact manifolds, and in particular Legendrian knots in with the standard contact structure. It is a classical result of [Bennequin] that is bounded above for Legendrian knots in any given topological knot type in . The maximal Thurston-Bennequin number of a smooth knot is the largest value of 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 , one can associate two integers: the writhe , defined as for any diagram by counting the number of crossings with signs ( for **Failed to parse (unknown function "\overcrossing"): {\displaystyle (\overcrossing)}**
and for **Failed to parse (unknown function "\undercrossing"): {\displaystyle (\undercrossing)}**
), and the cusp number , defined to be the number of locally upper-right corners of . Next define the Thurston-Bennequin number to be . Finally, the maximal Thurston-Bennequin number of a knot is the maximal value of over all rectilinear front diagrams in the knot type.

For example, the rectilinear front diagram in the figure, which is a right-handed trefoil, has , , and . In fact, the maximal Thurston-Bennequin number of the right-handed trefoil is .

In the Knot Atlas, maximal Thurston-Bennequin number is given as , where and 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).

[Bennequin] **^** D. Bennequin, *Entrelacements et équations de Pfaff*, Astérisque **107-108** (1983) 87-161.