Maximal Thurston-Bennequin number

2008-12-16T17:40:44Z

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

[[Image:RHtrefoil-rectilinear.gif|center]]

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

In the Knot Atlas, maximal Thurston-Bennequin number is given as <math>[a][b]</math>, where <math>a</math> and <math>b</math> are the maximal Thurston-Bennequin numbers of the knot and its mirror, respectively. The data has been imported from the KnotInfo site (see [http://www.indiana.edu/~knotinfo/descriptions/thurston_bennequin_number.html their page on the Thurston-Bennequin number]).

{{note|Bennequin}} D. Bennequin, ''Entrelacements et équations de Pfaff'', Astérisque '''107-108''' (1983) 87-161.

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Maximal Thurston-Bennequin number