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Link concordance

Link concordance 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 Link concordance rather than just read about it. In short: In mathematics, two links L 0 ⊂ S n {\displaystyle L_{0}\subset S^{n}} and L 1 ⊂ S n {\displaystyle L_{1}\subset S^{n}} are concordant if there exists an embedding f : L 0 × [ 0 , 1 ] → S n × [ 0 , 1 ] {\displaystyle f:L_{0}\times [0,1]\to S^{n}\times [0,1]} such that f ( L 0 × { 0 } ) = L 0 × { 0 } {\displaystyle f(L_{0}\times \{0\})=L_{0}\times \{0\}} and f ( L 0 × { 1 } ) = L 1 × { 1 } {\displaystyle f(L_{0}\time…

Key takeaways

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

Reference excerpt

In mathematics, two links L 0 ⊂ S n {\displaystyle L_{0}\subset S^{n}} and L 1 ⊂ S n {\displaystyle L_{1}\subset S^{n}} are concordant if there exists an embedding f : L 0 × [ 0 , 1 ] → S n × [ 0 , 1 ] {\displaystyle f:L_{0}\times [0,1]\to S^{n}\times [0,1]} such that f ( L 0 × { 0 } ) = L 0 × { 0 } {\displaystyle f(L_{0}\times \{0\})=L_{0}\times \{0\}} and f ( L 0 × { 1 } ) = L 1 × { 1 } {\displaystyle f(L_{0}\times \{1\})=L_{1}\times \{1\}} . By its nature, link concordance is an equivalence relation. It is weaker than isotopy, and stronger than homotopy: isotopy implies concordance implies homotopy. A link is a slice link if it is concordant to the unlink.

Concordance invariants A function of a link that is invariant under concordance is called a concordance invariant. The linking number of any two components of a link is one of the most elementary concordance invariants. The signature of a knot is also a concordance invariant. A subtler concordance invariant are the Milnor invariants, and in fact all rational finite type concordance invariants are Milnor invariants and their products, though non-finite type concordance invariants exist.

Higher dimensions One can analogously define concordance for any two submanifolds M 0 , M 1 ⊂ N {\displaystyle M_{0},M_{1}\subset N} . In this case one considers two submanifolds concordant if there is a cobordism between them in N × [ 0 , 1 ] , {\displaystyle N\times [0,1],} i.e., if there is a manifold with boundary W ⊂ N × [ 0 , 1 ] {\displaystyle W\subset N\times [0,1]} whose boundary consists of M 0 × { 0 } {\displaystyle M_{0}\times \{0\}} and M 1 × { 1 } . {\displaystyle M_{1}\times \{1\}.}

This higher-dimensional concordance is a relative form of cobordism – it requires two submanifolds to be not just abstractly cobordant, but "cobordant in N".

See also Slice knot

References

Further reading J. Hillman, Algebraic invariants of links. Series on Knots and everything. Vol 32. World Scientific. Livingston, Charles, A survey of classical knot concordance, in: Handbook of knot theory, pp 319–347, Elsevier, Amsterdam, 2005. MR 2179265 ISBN 0-444-51452-X

Worked examples

Example 1 — a first encounter with Link concordance

Start with the simplest possible case. Write down what Link concordance 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 Link concordance 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 Link concordance 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 Link concordance

In research
Link concordance 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 Link concordance 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
Link concordance is common in secondary-school and first-year university syllabi. It links to neighbouring topics Knot invariants, Manifolds, so understanding it makes those chapters shorter.
In everyday life
Look for Link concordance 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 Link concordance in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Link concordance 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 Link concordance out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Link concordance in simple terms?

In mathematics, two links L 0 ⊂ S n {\displaystyle L_{0}\subset S^{n}} and L 1 ⊂ S n {\displaystyle L_{1}\subset S^{n}} are concordant if there exists an embedding f : L 0 × [ 0 , 1 ] → S n × [ 0 , 1 ] {\displaystyle f:L_{0}\times [0,1]\to S^{n}\times [0,1]} such that f ( L 0 × { 0 } ) = L 0 × { 0…

Why does Link concordance 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 Link concordance?

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 Link concordance.

Tags

  • Knot invariants
  • Manifolds

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