ArticleslgStudy

science

Knot polynomial

Knot 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 Knot polynomial rather than just read about it. In short: In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot. History The first knot polynomial, the Alexander polynomial, was introduced by James Waddell Alexander II in 1923.

Knot polynomial — main illustration
Knot polynomial — illustration

Key takeaways

  • Knot 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 Knot polynomial to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Knot polynomial from memory before moving on to harder problems.

Reference excerpt

In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot.

History The first knot polynomial, the Alexander polynomial, was introduced by James Waddell Alexander II in 1923. Other knot polynomials were not found until almost 60 years later. In the 1960s, John Conway came up with a skein relation for a version of the Alexander polynomial, usually referred to as the Alexander–Conway polynomial. The significance of this skein relation was not realized until the early 1980s, when Vaughan Jones discovered the Jones polynomial. This led to the discovery of more knot polynomials, such as the so-called HOMFLY polynomial. Soon after Jones' discovery, Louis Kauffman noticed the Jones polynomial could be computed by means of a partition function (state-sum model), which involved the bracket polynomial, an invariant of framed knots. This opened up avenues of research linking knot theory and statistical mechanics. In the late 1980s, two related breakthroughs were made. Edward Witten demonstrated that the Jones polynomial, and similar Jones-type invariants, had an interpretation in Chern–Simons theory. Viktor Vasilyev and Mikhail Goussarov started the theory of finite type invariants of knots. The coefficients of the previously named polynomials are known to be of finite type (after perhaps a suitable "change of variables"). In recent years, the Alexander polynomial has been shown to be related to Floer homology. The graded Euler characteristic of the knot Floer homology of Peter Ozsváth and Zoltan Szabó is the Alexander polynomial.

Examples

Alexander–Briggs notation organizes knots by their crossing number. Alexander polynomials and Conway polynomials can not recognize the difference of left-trefoil knot and right-trefoil knot, while the Jones polynomial can.

So we have the same situation as the granny knot and square knot since the addition of knots in R 3 {\displaystyle \mathbb {R} ^{3}} is the product of knots in knot polynomials.

See also

Specific knot polynomials Alexander polynomial (and its variant, the Alexander-Conway polynomial) Bracket polynomial HOMFLY polynomial Jones polynomial Kauffman polynomial

Related topics Graph polynomial, a similar class of polynomial invariants in graph theory Tutte polynomial, a special type of graph polynomial related to the Jones polynomial Skein relation for a formal definition of the Alexander polynomial, with a worked-out example.

Further reading Adams, Colin. The Knot Book. American Mathematical Society. ISBN 0-8050-7380-9. Lickorish, W. B. R. (1997). An Introduction to Knot Theory. Graduate Texts in Mathematics. Vol. 175. New York: Springer-Verlag. ISBN 0-387-98254-X.

Illustrations

Knot polynomial: Many knot polynomials are computed using skein relations, which allow one to change the different crossings of a knot to get simpler knots.
Many knot polynomials are computed using skein relations, which allow one to change the different crossings of a knot to get simpler knots.
Knot polynomial illustration
Knot polynomial illustration

Worked examples

Example 1 — a first encounter with Knot polynomial

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

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

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Knot polynomial in 20 minutes

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

Frequently asked questions

What is Knot polynomial in simple terms?

In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot. History The first knot polynomial, the Alexander polynomial, was introduced by James Waddell Alexander II in 1923.

Why does Knot 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 Knot 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 Knot polynomial.

Tags

  • Knot invariants
  • Polynomials

Keep exploring