The Thurston Bennequin Number


Every knot has a Legendrian representative and the Thurston-Bennequin number of such a representative is defined using that Legendrian structure. The Thurston-Bennequin number of a knot is the maximum value of that invariant, taken over all possible Legendrian representatives. There is also a combinatorial definition. Every knot has a diagram such that at each crossing one strand intersects the bottom right and top left quadrants (formed by the vertical and horizontal lines at the crossing) and the other intersects the bottom left and top right quadrant. Furthermore it can be arranged that the strand that meets the bottom right quadrant passes over the other strand. For such a diagram, the Thurston-Bennequin number is the wirthe minus the number of right cusps (maximum points with respect to projection onto the x-axis. The Thurston-Bennequin number is the maximum value of this, taken over all diagrams satisfying the crossing criteria.

The Thurston-Bennequin number of a knot and its mirror image may be different, and in the table we list the known values for both possible orientations. The reader is referred to Ng's paper for a description of the correspondence between the to two values and the two choices of orientation.

In the table we list for each knot a pair of integers, the value of the invariant for a knot and for its mirror image. Thus, for 31 we list {-6}{1}: The negative trefoil has TB number -6, the positive trefoil has TB number 1.

The data for knots of 9 or few crossings was taken from a paper by Lenhard Ng. That paper leaves only one gap, the knot 942, with one of its two possible orientations.

In the reference "A Legendrian Thurston-Bennequin bound from Khovanov homology" Ng has developed new bounds of the Thurston Bennequin number. Using these he provided us with the latest information for 9 and 10 crossing number knots.

Recent work of Ng and Baldwin-Gilliam have provided compuations of the Thurston-Benniquin numbers of most knots through 11 crossings.


References

J. Baldwin and W. Gillam, Computations of Heegaard-Floer knot homology, math.GT/0610167.

L. Ng, Maximal Thurston-Bennequin Number of Two-Bridge Links Algebr, Geom. Topol. 1 (2001) 427-434.

L. Ng, A Legendrian Thurston-Bennequin bound from Khovanov homology, Arxiv:math.GT/0508649

L. Ng, On Arc index and maximal Thurston-Bennequin number, math.GT/0612356.

Ivan Dynnikov and Maxim Prasolov have shown that tb(K) + tb(-K) = arc index (K): see ArXiv preprint. Furthermore, they prove that the maximal Thurston-Bennequin number is realized by a diagram with minimal arc index.


Further information on particular knots.