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Legendrian knot

Legendrian knot is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Legendrian knot rather than just read about it. In short: In mathematics, a Legendrian knot often refers to a smooth embedding of the circle into R 3 {\displaystyle \mathbb {R} ^{3}} , which is tangent to the standard contact structure on R 3 {\displaystyle \mathbb {R} ^{3}} . It is the lowest-dimensional case of a Legendrian submanifold, which is an embedding of a k-dimensional manifold into a (2k+1)-dimensional contact manifold that is always tangent to the contact hyper…

Legendrian knot — main illustration
Legendrian knot — illustration

Key takeaways

  • Legendrian knot belongs to science; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Legendrian knot to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Legendrian knot from memory before moving on to harder problems.

Reference excerpt

In mathematics, a Legendrian knot often refers to a smooth embedding of the circle into R 3 {\displaystyle \mathbb {R} ^{3}} , which is tangent to the standard contact structure on R 3 {\displaystyle \mathbb {R} ^{3}} . It is the lowest-dimensional case of a Legendrian submanifold, which is an embedding of a k-dimensional manifold into a (2k+1)-dimensional contact manifold that is always tangent to the contact hyperplane.

Classification Two smooth knots are equivalent if there is a way to smoothly deform one into the other. That is, if there is a smooth ambient isotopy from one to the other. Similarly, two Legendrian knots are equivalent if there is a way to smoothly deform one into the other, such that any intermediate knot is still a Legendrian knot. Two equivalent Legendrian knots are equivalent as smooth knots, but the converse is false. Many inequivalent Legendrian knots can be distinguished by considering their Thurston-Bennequin invariants and rotation number, which are together known as the "classical invariants" of Legendrian knots. More sophisticated invariants have been constructed, including one constructed combinatorially by Chekanov and using holomorphic discs by Eliashberg. This Chekanov-Eliashberg invariant yields an invariant for loops of Legendrian knots by considering the monodromy of the loops. This has yielded noncontractible loops of Legendrian knots which are contractible in the space of all knots. Any Legendrian knot may be C 0 {\displaystyle C^{0}} perturbed to a transverse knot (a knot transverse to a contact structure) by pushing off in a direction transverse to the contact planes. The set of isomorphism classes of Legendrian knots modulo negative Legendrian stabilizations is in bijection with the set of transverse knots.

References

Geiges, Hansjörg (2008). An introduction to contact topology; Volume 109 of Cambridge studies in advanced mathematics. Cambridge University Press. p. 94. ISBN 978-0-521-86585-2. Casacuberta, Carlos (2001). European Congress of Mathematics: Barcelona, July 10–14, 2000. Birkhäuser. p. 526. ISBN 978-3764364182. Epstein, J.; Fuchs, D.; Meyer, M. (2001). "Chekanov–Eliashberg invariants and transverse approximations of Legendrian knots". Pacific Journal of Mathematics. 201 (1): 89–106. doi:10.2140/pjm.2001.201.89. Kalman, Tamas (2005). "Contact homology and one parameter families of Legendrian knots". Geometry & Topology. 9 (4): 2013–2078. arXiv:math/0407347. doi:10.2140/gt.2005.9.2013. S2CID 8307055. Sabloff, Joshua M. (2009), "What Is . . . a Legendrian Knot?" (PDF), AMS Notices, 56 (10): 1282–1284.

External links The Legendrian knot atlas

Illustrations

Legendrian knot: The standard contact structure on R3. Each point in R3 has a plane associated to it by the contact structure, in this case as the kernel of the one-form dz − y dx.
The standard contact structure on R3. Each point in R3 has a plane associated to it by the contact structure, in this case as the kernel of the one-form dz − y dx.

Worked examples

Example 1 — a first encounter with Legendrian knot

Start with the simplest possible case. Write down what Legendrian knot claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Legendrian knot before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Legendrian knot ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Legendrian knot

In research
Legendrian knot appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Legendrian knot in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Legendrian knot is common in secondary-school and first-year university syllabi. It links to neighbouring topics Knot theory stubs, Knots and links, so understanding it makes those chapters shorter.
In everyday life
Look for Legendrian knot outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Legendrian knot in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Legendrian knot means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Legendrian knot out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Legendrian knot in simple terms?

In mathematics, a Legendrian knot often refers to a smooth embedding of the circle into R 3 {\displaystyle \mathbb {R} ^{3}} , which is tangent to the standard contact structure on R 3 {\displaystyle \mathbb {R} ^{3}} . It is the lowest-dimensional case of a Legendrian submanifold, which is an embe…

Why does Legendrian knot matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Legendrian knot?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Legendrian knot.

Tags

  • Knot theory stubs
  • Knots and links

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