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 grid diagram, as illustrated for the knot 6_2 below. Such a diagram consists of a collection of vertical and horizontal intervals with vertices at lattice points. The vertices associated to the diagram of 6_2 are

knot diagram

Note that in this example, on the 8 by 8 grid there are exactly 2 vertices for each possible value of the first and second coordinates. Note also that the vertical segments always pass over the horizontal segments. These are defining proporties of grid diagrams. See Grid Notation for more information. Any grid diagram represents a Legendrian knot k, and its associated Thurston-Bennequin number "tb(k)" is computed as the writhe of the diagram minus the number of northwest corners. For a given knot K, the maximum of tb(k) for all Legendrian representatives k is called the Thurston-Bennequin number of K. It is denoted TB(K). Dynnikor and Prasolov[2] proved that the maximum Thurston-Bennequin number of a knot, TB(K) is realized as tb(k) for the Legendrian knot with the smallest grid number. In particular, the computation of grid numbers and Thurston Bennequin numbers is a finite computational problem, though apparently of exponential complexity.

The Thurston-Bennequin number of a knot and its mirror image may be different. A grid diagram for the mirror image of a knot can be obtained by rotating the diagram by 90 degrees (while switching crossings so that verticals continue to be over horizontal segments). In the table, we list the known values for both possible orientations of each knot. Thus, for 31 we list {1}{-6}: The positive trefoil has TB number 1, the negative trefoil has TB number -6. In the database, the first TB number corresponds to the knot defined by the PD notation given in KnotInfo as illustrated in the various diagrams of the knot.

See Grid Notation for up-to-date information, including the work of Gyo Taek Jin and coauthors. Here is a summary of the work that was used for the initial data in KnotInfo.

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.

Rick Litherland provided us with Thurston-Bennequin numbers for most 12 crossing prime knots. Results of Drynnikov-Prasolov [2] led to the completion of the computation for 12 crossings.

Specific Knots

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].


[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.

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