This is a gallery of some Legendrian knots in R3 with the standard contact structure. The kernel of the one-form dz - y dx defines a plane at every point in R3, and a Legendrian knot is a knot that is everywhere tangent to this plane field.
For each knot, several different perspectives of the same physical knot are given. Each knot is slightly thickened for clarity, and the contact plane is drawn at various points along the knot. All pictures were rendered by me using Mathematica.
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Unknot |
Thurston-Bennequin
number -1 |
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Stabilized unknot |
Thurston-Bennequin number -2 |
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Right-handed trefoil |
Thurston-Bennequin number 1 |
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Left-handed trefoil |
Thurston-Bennequin number -6 |
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Chekanov 52 knot A
(listed as the first m(52) knot
in the Legendrian knot atlas) |
Thurston-Bennequin number 1 |
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Chekanov 52 knot B
(listed as the second m(52) knot
in the Legendrian knot atlas) |
Thurston-Bennequin number 1 |
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