Maximal Thurston-Bennequin number: Difference between revisions
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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. |
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. |
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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>. |
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>. |
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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. |
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]). |
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{{note|Bennequin}} D. Bennequin, ''Entrelacements et équations de Pfaff'', Astérisque '''107-108''' (1983) 87-161. |
{{note|Bennequin}} D. Bennequin, ''Entrelacements et équations de Pfaff'', Astérisque '''107-108''' (1983) 87-161. |
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Latest revision as of 07:45, 17 December 2008
The Thurston-Bennequin number, usually denoted [math]\displaystyle{ tb }[/math], is an invariant of nullhomologous Legendrian knots in contact manifolds, and in particular Legendrian knots in [math]\displaystyle{ {\mathbf R}^3 }[/math] with the standard contact structure. It is a classical result of [Bennequin] that [math]\displaystyle{ tb }[/math] is bounded above for Legendrian knots in any given topological knot type in [math]\displaystyle{ {\mathbf R}^3 }[/math]. The maximal Thurston-Bennequin number of a smooth knot is the largest value of [math]\displaystyle{ 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]\displaystyle{ F }[/math], one can associate two integers: the writhe [math]\displaystyle{ w(F) }[/math], defined as for any diagram by counting the number of crossings with signs ([math]\displaystyle{ +1 }[/math] for [math]\displaystyle{ (\overcrossing) }[/math] and [math]\displaystyle{ -1 }[/math] for [math]\displaystyle{ (\undercrossing) }[/math]), and the cusp number [math]\displaystyle{ c(F) }[/math], defined to be the number of locally upper-right corners of [math]\displaystyle{ F }[/math]. Next define the Thurston-Bennequin number [math]\displaystyle{ tb(F) }[/math] to be [math]\displaystyle{ w(F)-c(F) }[/math]. Finally, the maximal Thurston-Bennequin number of a knot is the maximal value of [math]\displaystyle{ tb(F) }[/math] over all rectilinear front diagrams [math]\displaystyle{ F }[/math] in the knot type.
For example, the rectilinear front diagram in the figure, which is a right-handed trefoil, has [math]\displaystyle{ w=3 }[/math], [math]\displaystyle{ c=2 }[/math], and [math]\displaystyle{ tb=1 }[/math]. In fact, the maximal Thurston-Bennequin number of the right-handed trefoil is [math]\displaystyle{ 1 }[/math].
In the Knot Atlas, maximal Thurston-Bennequin number is given as [math]\displaystyle{ [a][b] }[/math], where [math]\displaystyle{ a }[/math] and [math]\displaystyle{ 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 their page on the Thurston-Bennequin number).
[Bennequin] ^ D. Bennequin, Entrelacements et équations de Pfaff, Astérisque 107-108 (1983) 87-161.
