mathematics

Meyerhoff manifold

Meyerhoff manifold 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 Meyerhoff manifold rather than just read about it. In short: In hyperbolic geometry, the Meyerhoff manifold is the arithmetic hyperbolic 3-manifold obtained by ( 5 , 1 ) {\displaystyle (5,1)} surgery on the figure-8 knot complement. It was introduced by Robert Meyerhoff (1987) as a possible candidate for the hyperbolic 3-manifold of smallest volume, but the Weeks manifold turned out to have slightly smaller volume.

Key takeaways

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

Reference excerpt

In hyperbolic geometry, the Meyerhoff manifold is the arithmetic hyperbolic 3-manifold obtained by ( 5 , 1 ) {\displaystyle (5,1)} surgery on the figure-8 knot complement. It was introduced by Robert Meyerhoff (1987) as a possible candidate for the hyperbolic 3-manifold of smallest volume, but the Weeks manifold turned out to have slightly smaller volume. It has the second smallest volume

V m = 12 ⋅ ( 283 ) 3 / 2 ζ k ( 2 ) ( 2 π ) − 6 = 0.981368 … {\displaystyle V_{m}=12\cdot (283)^{3/2}\zeta _{k}(2)(2\pi )^{-6}=0.981368\dots }

of orientable arithmetic hyperbolic 3-manifolds, where ζ k {\displaystyle \zeta _{k}} is the zeta function of the quartic field of discriminant − 283 {\displaystyle -283} . Alternatively,

V m = ℑ ( L i 2 ( θ ) + ln ⁡ | θ | ln ⁡ ( 1 − θ ) ) = 0.981368 … {\displaystyle V_{m}=\Im ({\rm {{Li}_{2}(\theta )+\ln |\theta |\ln(1-\theta ))=0.981368\dots }}}

where L i n {\displaystyle {\rm {{Li}_{n}}}} is the polylogarithm and | x | {\displaystyle |x|} is the absolute value of the complex root θ {\displaystyle \theta } (with positive imaginary part) of the quartic θ 4 + θ − 1 = 0 {\displaystyle \theta ^{4}+\theta -1=0} . Ted Chinburg (1987) showed that this manifold is arithmetic.

See also Gieseking manifold Weeks manifold

References Chinburg, Ted (1987), "A small arithmetic hyperbolic three-manifold", Proceedings of the American Mathematical Society, 100 (1): 140–144, doi:10.2307/2046135, ISSN 0002-9939, JSTOR 2046135, MR 0883417 Chinburg, Ted; Friedman, Eduardo; Jones, Kerry N.; Reid, Alan W. (2001), "The arithmetic hyperbolic 3-manifold of smallest volume", Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV, 30 (1): 1–40, ISSN 0391-173X, MR 1882023 Meyerhoff, Robert (1987), "A lower bound for the volume of hyperbolic 3-manifolds", Canadian Journal of Mathematics, 39 (5): 1038–1056, doi:10.4153/CJM-1987-053-6, ISSN 0008-414X, MR 0918586

Worked examples

Example 1 — a first encounter with Meyerhoff manifold

Start with the simplest possible case. Write down what Meyerhoff manifold 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 Meyerhoff manifold 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 Meyerhoff manifold 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 Meyerhoff manifold

In research: Meyerhoff manifold 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 Meyerhoff manifold 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: Meyerhoff manifold is common in secondary-school and first-year university syllabi. It links to neighbouring topics 3-manifolds, Hyperbolic manifolds, Metric geometry stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Meyerhoff manifold 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 Meyerhoff manifold in 20 minutes

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

Frequently asked questions

What is Meyerhoff manifold in simple terms?

Why does Meyerhoff manifold 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 Meyerhoff manifold?

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 Meyerhoff manifold.