Weeks manifold

Key takeaways

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

Reference excerpt

In mathematics, the Weeks manifold, sometimes called the Fomenko–Matveev–Weeks manifold, is a closed hyperbolic 3-manifold obtained by (5, 2) and (5, 1) Dehn surgeries on the Whitehead link. It has volume approximately equal to 0.942707… (OEIS: A126774) and David Gabai, Robert Meyerhoff, and Peter Milley (2009) showed that it has the smallest volume of any closed orientable hyperbolic 3-manifold. The manifold was independently discovered by Jeffrey Weeks (1985) as well as Sergei V. Matveev and Anatoly T. Fomenko (1988).

Volume Since the Weeks manifold is an arithmetic hyperbolic 3-manifold, its volume can be computed using its arithmetic data and a formula due to Armand Borel:

V w = 3 ⋅ 23 3 / 2 ζ k ( 2 ) 4 π 4 = 0.942707 … {\displaystyle V_{w}={\frac {3\cdot 23^{3/2}\zeta _{k}(2)}{4\pi ^{4}}}=0.942707\dots }

where k {\displaystyle k} is the number field generated by θ {\displaystyle \theta } satisfying θ 3 − θ + 1 = 0 {\displaystyle \theta ^{3}-\theta +1=0} and ζ k {\displaystyle \zeta _{k}} is the Dedekind zeta function of k {\displaystyle k} . Alternatively,

V w = ℑ ( L i 2 ( θ ) + ln ⁡ | θ | ln ⁡ ( 1 − θ ) ) = 0.942707 … {\displaystyle V_{w}=\Im ({\rm {{Li}_{2}(\theta )+\ln |\theta |\ln(1-\theta ))=0.942707\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 cubic.

Symmetries The Weeks manifold has symmetry group D 6 {\displaystyle D_{6}} , the dihedral group of order 12. Quotients by this group and its subgroups can be used to characterize the manifold as a branched covering based on an orbifold. In particular, the quotient by the order-3 subgroup of the symmetry group has underlying set a 3-sphere and branch set a 52 knot.

Related manifolds The cusped hyperbolic 3-manifold obtained by (5, 1) Dehn surgery on the Whitehead link is the so-called sibling manifold, or sister, of the figure-eight knot complement. The figure eight knot's complement and its sibling have the smallest volume of any orientable, cusped hyperbolic 3-manifold. Thus the Weeks manifold can be obtained by hyperbolic Dehn surgery on one of the two smallest orientable cusped hyperbolic 3-manifolds.

See also Meyerhoff manifold – another manifold with very small volume

References

Agol, Ian; Storm, Peter A.; Thurston, William P. (2007), "Lower bounds on volumes of hyperbolic Haken 3-manifolds (with an appendix by Nathan Dunfield)", Journal of the American Mathematical Society, 20 (4): 1053–1077, arXiv:math.DG/0506338, Bibcode:2007JAMS...20.1053A, doi:10.1090/S0894-0347-07-00564-4, MR 2328715. 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, MR 1882023 Gabai, David; Meyerhoff, Robert; Milley, Peter (2009), "Minimum volume cusped hyperbolic three-manifolds", Journal of the American Mathematical Society, 22 (4): 1157–1215, arXiv:0705.4325, Bibcode:2009JAMS...22.1157G, doi:10.1090/S0894-0347-09-00639-0, MR 2525782 Matveev, Sergei V.; Fomenko, Aanatoly T. (1988), "Isoenergetic surfaces of Hamiltonian systems, the enumeration of three-dimensional manifolds in order of growth of their complexity, and the calculation of the volumes of closed hyperbolic manifolds", Russian Mathematical Surveys, 43 (1): 3–24, Bibcode:1988RuMaS..43....3M, doi:10.1070/RM1988v043n01ABEH001554, MR 0937017 Weeks, Jeffrey (1985), Hyperbolic structures on 3-manifolds, Ph.D. thesis, Princeton University

Worked examples

Example 1 — a first encounter with Weeks manifold

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

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

How to study Weeks manifold in 20 minutes

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

Frequently asked questions

What is Weeks manifold in simple terms?

In mathematics, the Weeks manifold, sometimes called the Fomenko–Matveev–Weeks manifold, is a closed hyperbolic 3-manifold obtained by (5, 2) and (5, 1) Dehn surgeries on the Whitehead link. It has volume approximately equal to 0.942707… (OEIS: A126774) and David Gabai, Robert Meyerhoff, and Peter…

Why does Weeks manifold 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 Weeks 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 Weeks manifold.