Maximal Thurston-Bennequin number

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


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

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

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