Maximal Thurston-Bennequin number

From Knot Atlas
Revision as of 07:45, 17 December 2008 by Drorbn (talk | contribs) (Reverted edits by ErermOnboo (Talk); changed back to last version by Drorbn)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigationJump to search

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


For example, the rectilinear front diagram in the figure, which is a right-handed trefoil, has , , and . In fact, the maximal Thurston-Bennequin number of the right-handed trefoil is .

In the Knot Atlas, maximal Thurston-Bennequin number is given as , where and 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.