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. The reader is referred to [?] for a description of the correspondence between the two values and the two choices of orientation. In the table, we list the known values for both possible orientations of each knot. Thus, for 31 we list {-6}{1}: The negative trefoil has TB number -6, the positive trefoil has TB number 1.

Specific Knots

The data for knots of 9 or fewer crossings was taken from a paper by Lenhard Ng. That paper leaves only one gap, 942, which was unknown for only one of the two orientations. It has now been resolved in [3].

In [3], Ng developed new bounds of the Thurston-Bennequin number. Using these, he provided us with the latest information for 9 and 10 crossing knots.

Recent work by Ng and Baldwin-Gilliam has provided computations of the Thurston-Benniquin numbers of most knots through 11 crossings.

Lenhard Ng, in [5], resolved the final cases for knots of 11 or fewer crossings.

12n_{41, 119, 120, 121, 145, 153, 199, 200, 243, 260, 282, 310, 322, 351, 362, 368, 377, 403, 414, 425, 475, 523, 549}
   Ivan Dynnikov and Maxim Prasolov have shown that tb(K) + tb(-K) = - arc index (K). This resolves the last cases of unknown Thurston-Bennequin numbers for these prime knots of 12 or fewer crossings. Furthermore, they prove that the maximal Thurston-Bennequin number is realized by a diagram with minimal arc index. See [2].

References

[1] Baldwin, J. and Gillam, W., "Computations of Heegaard-Floer knot homology", ArXiv preprint.

[2] Dynnikor, I. and Prasolov, M., "Bypasses for rectangular diagrams. Proof of Jones' conjecture and related questions," ArXiv preprint.

[3] Ng, L., "A Legendrian Thurston-Bennequin bound from Khovanov homology." Algebr. Geom. Topol. 5 (2005), 1637-1653.

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

[5] Ng, L., "On arc index and maximal Thurston-Benniquin Number," ArXiv preprint.