science

Knot tabulation

Knot tabulation 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 tabulation rather than just read about it. In short: Ever since Sir William Thomson's vortex theory, mathematicians have tried to classify and tabulate all possible knots. By 1998, all 1.7 million prime knots up to 16 crossings had been tabulated, and by 2020 all 350 million knots up to 19 crossings had been tabulated.

Key takeaways

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

Reference excerpt

Ever since Sir William Thomson's vortex theory, mathematicians have tried to classify and tabulate all possible knots. By 1998, all 1.7 million prime knots up to 16 crossings had been tabulated, and by 2020 all 350 million knots up to 19 crossings had been tabulated. The major challenge of the process is that many apparently different knots may actually be different geometrical presentations of the same topological entity, and that proving or disproving knot equivalence is much more difficult than it at first seems.

Beginnings

In the 19th century, Sir William Thomson made a hypothesis that the chemical elements were based upon knotted vortices in the aether. In an attempt to make a periodic table of the elements, P. G. Tait, C. N. Little and others started to attempt to count all possible knots. Because their work predated the invention of the digital computer, all work had to be done by hand.

Perko pair

In 1974, Kenneth Perko discovered a duplication in the Tait-Little tables, called the Perko pair. Later knot tables took two approaches to resolving this: some just skipped one of the entries without renumbering, and others renumbered the later entries to remove the hole. The resulting ambiguity has continued to the present day, and has been further compounded by mistaken attempts to correct errors caused by this that were themselves incorrect. For example, Wolfram Web's Perko Pair page erroneously compares two different knots (due to the renumbering by mathematicians such as Burde and Bar-Natan).

New methods Jim Hoste, Jeff Weeks, and Morwen Thistlethwaite used computer searches to count all knots with 16 or fewer crossings. This research was performed separately using two different algorithms on different computers, lending support to the correctness of its results. Both counts found 1701936 prime knots (including the unknot) with up to 16 crossings. Most recently, in 2020, Benjamin Burton classified all prime knots up to 19 crossings (of which there are almost 300 million). Starting with three crossings (the minimum for any nontrivial knot), the number of prime knots for each number of crossings is

1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, 46972, 253293, 1388705, ... (sequence A002863 in the OEIS) Modern automated methods can now enumerate billions of knots in a matter of days.

See also Knot theory Knot (mathematics) List of prime knots Unknotting problem

References

Illustrations

Worked examples

Example 1 — a first encounter with Knot tabulation

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

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

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

Frequently asked questions

What is Knot tabulation in simple terms?

Why does Knot tabulation 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 tabulation?

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 tabulation.