mathematics

Unknotting number

Unknotting number is a mathematics 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 Unknotting number rather than just read about it. In short: In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself (crossing switch) to untie it. If a knot has unknotting number n {\displaystyle n} , then there exists a diagram of the knot which can be changed to unknot by switching n {\displaystyle n} crossings.

Key takeaways

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

Reference excerpt

In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself (crossing switch) to untie it. If a knot has unknotting number n {\displaystyle n} , then there exists a diagram of the knot which can be changed to unknot by switching n {\displaystyle n} crossings. The unknotting number of a knot is always less than half of its crossing number. This invariant was first defined by Hilmar Wendt in 1936. Any composite knot has unknotting number at least two, and therefore every knot with unknotting number one is a prime knot. The unknotting number is not additive under connected sum, although that possibility, implicit in [Wendt,1937] and explicitly asked by Gordon in 1977 and many others, was not resolved until 2025. A counterexample showed that the unknotting number of the connected sum of 71 and its mirror image was at most 5, one less than the sum of the numbers from its components. The following table show the unknotting numbers for the first few knots:

In general, it is relatively difficult to determine the unknotting number of a given knot. Known cases include:

The unknotting number of a nontrivial twist knot is always equal to one. The unknotting number of a ( p , q ) {\displaystyle (p,q)} -torus knot is equal to ( p − 1 ) ( q − 1 ) / 2 {\displaystyle (p-1)(q-1)/2} . The unknotting numbers of prime knots with nine or fewer crossings have all been determined. (The unknotting number of the 1011 prime knot is unknown.)

Other numerical knot invariants Crossing number Bridge number Linking number Stick number

Illustrations

Worked examples

Example 1 — a first encounter with Unknotting number

Start with the simplest possible case. Write down what Unknotting number claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Unknotting number 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 Unknotting number 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 Unknotting number

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

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

Frequently asked questions

What is Unknotting number in simple terms?

Why does Unknotting number matter?

Because it connects several mathematics 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 Unknotting number?

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 Unknotting number.