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Jones polynomial

Jones polynomial 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 Jones polynomial rather than just read about it. In short: In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t 1 / 2 {\displaystyle t^{1/2}} with integer coefficients.

Jones polynomial — main illustration
Jones polynomial — illustration

Key takeaways

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

Reference excerpt

In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t 1 / 2 {\displaystyle t^{1/2}} with integer coefficients.

Definition by the bracket

Suppose we have an oriented link L {\displaystyle L} , given as a knot diagram. We will define the Jones polynomial V ( L ) {\displaystyle V(L)} by using Louis Kauffman's bracket polynomial, which we denote by ⟨ ⟩ {\displaystyle \langle ~\rangle } . Here the bracket polynomial is a Laurent polynomial in the variable A {\displaystyle A} with integer coefficients. First, we define the auxiliary polynomial (also known as the normalized bracket polynomial)

X ( L ) = ( − A 3 ) − w ( L ) ⟨ L ⟩ , {\displaystyle X(L)=(-A^{3})^{-w(L)}\langle L\rangle ,}

where w ( L ) {\displaystyle w(L)} denotes the writhe of L {\displaystyle L} in its given diagram. The writhe of a diagram is the number of positive crossings ( L + {\displaystyle L_{+}} in the figure below) minus the number of negative crossings ( L − {\displaystyle L_{-}} ). The writhe is not a knot invariant.

X ( L ) {\displaystyle X(L)} is a knot invariant since it is invariant under changes of the diagram of L {\displaystyle L} by the three Reidemeister moves. Invariance under type II and III Reidemeister moves follows from invariance of the bracket under those moves. The bracket polynomial is known to change by a factor of − A ± 3 {\displaystyle -A^{\pm 3}} under a type I Reidemeister move. The definition of the X {\displaystyle X} polynomial given above is designed to nullify this change, since the writhe changes appropriately by + 1 {\displaystyle +1} or − 1 {\displaystyle -1} under type I moves. Now make the substitution A = t − 1 / 4 {\displaystyle A=t^{-1/4}} in X ( L ) {\displaystyle X(L)} to get the Jones polynomial V ( L ) {\displaystyle V(L)} . This results in a Laurent polynomial with integer coefficients in the variable t 1 / 2 {\displaystyle t^{1/2}} .

Jones polynomial for tangles This construction of the Jones polynomial for tangles is a simple generalization of the Kauffman bracket of a link. The construction was developed by Vladimir Turaev and published in 1990. Let k {\displaystyle k} be a non-negative integer and S k {\displaystyle S_{k}} denote the set of all isotopic types of tangle diagrams, with 2 k {\displaystyle 2k} ends, having no crossing points and no closed components (smoothings). Turaev's construction makes use of the previous construction for the Kauffman bracket and associates to each 2 k {\displaystyle 2k} -end oriented tangle an element of the free R {\displaystyle \mathrm {R} } -module R [ S k ] {\displaystyle \mathrm {R} [S_{k}]} , where R {\displaystyle \mathrm {R} } is the ring of Laurent polynomials with integer coefficients in the variable t 1 / 2 {\displaystyle t^{1/2}} .

… excerpt ends here. Continue reading the full article.

Illustrations

Jones polynomial illustration
Jones polynomial: The simplest link with the same Jones polynomial as the unlink. The black and red components are a trefoil and figure-eight knot respectively.
The simplest link with the same Jones polynomial as the unlink. The black and red components are a trefoil and figure-eight knot respectively.

Worked examples

Example 1 — a first encounter with Jones polynomial

Start with the simplest possible case. Write down what Jones polynomial 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 Jones polynomial 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 Jones polynomial 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 Jones polynomial

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

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

Frequently asked questions

What is Jones polynomial in simple terms?

In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t 1 / 2 {\displaystyle t^{1/2}} with…

Why does Jones polynomial 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 Jones polynomial?

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 Jones polynomial.

Tags

  • Knot invariants
  • Knot theory
  • Polynomials

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