Maximal Thurston-Bennequin number

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The Thurston-Bennequin number, usually denoted [math]\displaystyle{ tb }[/math], is an invariant of nullhomologous Legendrian knots in contact manifolds, and in particular Legendrian knots in [math]\displaystyle{ {\mathbf R}^3 }[/math] with the standard contact structure. It is a classical result of [Bennequin] that [math]\displaystyle{ tb }[/math] is bounded above for Legendrian knots in any given topological knot type in [math]\displaystyle{ {\mathbf R}^3 }[/math]. The maximal Thurston-Bennequin number of a smooth knot is the largest value of [math]\displaystyle{ tb }[/math] 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 [math]\displaystyle{ F }[/math], one can associate two integers: the writhe [math]\displaystyle{ w(F) }[/math], defined as for any diagram by counting the number of crossings with signs ([math]\displaystyle{ +1 }[/math] for [math]\displaystyle{ (\overcrossing) }[/math] and [math]\displaystyle{ -1 }[/math] for [math]\displaystyle{ (\undercrossing) }[/math]), and the cusp number [math]\displaystyle{ c(F) }[/math], defined to be the number of locally upper-right corners of [math]\displaystyle{ F }[/math]. Next define the Thurston-Bennequin number [math]\displaystyle{ tb(F) }[/math] to be [math]\displaystyle{ w(F)-c(F) }[/math]. Finally, the maximal Thurston-Bennequin number of a knot is the maximal value of [math]\displaystyle{ tb(F) }[/math] over all rectilinear front diagrams [math]\displaystyle{ F }[/math] in the knot type.

For example, the rectilinear front diagram in the figure, which is a right-handed trefoil, has [math]\displaystyle{ w=3 }[/math], [math]\displaystyle{ c=2 }[/math], and [math]\displaystyle{ tb=1 }[/math]. In fact, the maximal Thurston-Bennequin number of the right-handed trefoil is [math]\displaystyle{ 1 }[/math].

In the Knot Atlas, maximal Thurston-Bennequin number is given as [math]\displaystyle{ [a][b] }[/math], where [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] are the maximal Thurston-Bennequin numbers of the knot and its mirror, respectively.

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