A somewhat more sophisticated example of a space with a non-abelian fundamental group is the complement of a trefoil knot in R < sup > 3 </ sup >.

The independent and distinct approaches of Robert Riley and Troels Jørgensen in the mid-to-late 1970s showed that such examples were less atypical than previously believed ; in particular their work showed that the figure eight knot complement was hyperbolic.

Inspired by their work, Thurston took a different, more explicit means of exhibiting the hyperbolic structure of the figure eight knot complement.

He showed that the figure eight knot complement could be decomposed as the union of two regular ideal hyperbolic tetrahedra whose hyperbolic structures matched up correctly and gave the hyperbolic structure on the figure eight knot complement.

By utilizing Haken's normal surface techniques, he classified the incompressible surfaces in the knot complement.

The figure-eight knot is also the hyperbolic knot whose complement has the smallest possible volume, 2. 02988 ... according to the work of Chun Cao and Robert Meyerhoff.

The figure eight knot complement is a double-cover of the Gieseking manifold, which has the smallest volume among non-compact hyperbolic 3-manifolds.

The knot group of the unknot is an infinite cyclic group, and the knot complement is homeomorphic to a solid torus.

Some invariants associated with the knot complement include the knot group which is just the fundamental group of the complement.

If the manifold is non-compact, then the fundamental group cannot distinguish the two geometries, and there are examples ( such as the complement of a trefoil knot ) where a manifold may have a finite volume geometric structure of either type.

Given a knot in the 3-sphere, the knot complement is all the points of the 3-sphere not contained in the knot.

A knot whose complement has a non-trivial JSJ decomposition.

When go represents itself and a complement ( being equivalent, say, to go to Martinique ) in which boat did Jack go on??

when it represents only itself and on which is its complement ( so that go on is semantically equivalent to board ), on has stronger stress than go does.

Thus if an event or its complement almost surely occurs, it is independent of itself.

The definition is that a σ-algebra over a set is a nonempty collection of subsets of ( including itself ) that is closed under the complement and countable unions of its members.

The clock tower is designed on an axis with, and to complement, the cupola on top of the mansion itself.

Larger birds ( e. g. sandpipers, flickers and even Rock Pigeons as heavy as the Merlin itself ) and other animals — insects ( especially dragonflies and moths ), small mammals ( especially bats and voles ) and reptiles — complement its diet.

A set is defined to be closed if its complement is open, which leaves the possibility of an open set whose complement is itself also open, making the first set both open and closed, and therefore clopen.

Philips named the videotape standard Video Compact Cassette ( VCC ) to complement their landmark Audio Compact Cassette format introduced in 1963, but the format itself was marketed under the trademark Video 2000.

In some versions of the Minimalist Program the DP is itself the complement of a phase head, n *, from which it inherits the ability to agree with its complement and assign case.

By itself, IgM is an ineffective opsonin ; however it contributes greatly to opsonization by activating complement and causing C3b to bind to the antigen.

BUA's bid to become the new " Second Force " centred on a six-year expansion plan that envisaged the progressive transfer of all of BOAC's African routes to itself to complement existing African services from the airline's Gatwick base, and unlimited frequencies between Gatwick and New York's John F. Kennedy Airport.

Unusually, the MTA has placed identifying signage on concrete pilings opposite the platform to complement the signs on the platform itself.

This led to restoration of the affected rooms, as well as being the impetus for an extensive redevelopment of the property through the construction of barns and outbuildings to complement the house itself and to accommodate visitor interpretation facilities.

Therefore, the complement of any comparability graph is perfect ; this is essentially just Dilworth's theorem itself, restated in graph-theoretic terms.

To see this, note that for every subspace of a Hilbert space, there exists a unique linear transformation from the Hilbert space to itself which maps points on the subspace to itself while mapping points on its orthogonal complement to zero.

As a punishment, it differs from a fine in that it is not primarily meant to match the crime but rather reattributes the criminal's ill-gotten spoils ( often as a complement to the actual punishment for the crime itself ; still common with various kinds of contraband, such as protected living organisms ) to the community or even aims to rob them of their socio-economic status, in the extreme case reducing them to utter poverty, or if he is condemned to death even denies them inheritance to the legal heirs.

In this analysis, it is usually a reduced complement factor C4, rather than the C1-INH deficiency itself, that is detected.

But the definition itself is adequate and does not need complement of commentary.

Ward's persuasion on the question of female intelligence as described by himself: " And now from the point of view of intellectual development itself we find her side by side, and shoulder to shoulder with him furnishing, from the very outset, far back in prehistoric, presocial, and even prehuman times, the necessary complement to his otherwise one-sided, headlong, and wayward career, without which he would soon have warped and distorted the race and rendered it incapable of the very progress which he claims exclusively to inspire.

The complement operation is an involution, meaning it " undoes itself ", or the complement of the complement is the original problem.

In mathematics, a Borel set is any set in a topological space that can be formed from open sets ( or, equivalently, from closed sets ) through the operations of countable union, countable intersection, and relative complement.

In general, if E is any subset of a topological space X such that both E and the complement of E are dense in X, then the real-valued function which takes the value 1 on E and 0 on the complement of E will be nowhere continuous.

Topology inherits a duality between open and closed subsets of some fixed topological space X: a subset U of X is closed if and only if its complement in X is open.

Under the name Riemann's existence theorem a deeper result on ramified coverings of a compact Riemann surface was known: such finite coverings as topological spaces are classified by permutation representations of the fundamental group of the complement of the ramification points.

Stated in terms of topology alone, a topological space X has a fork if X has a closed subset T with connected interior, whose boundary consists of three distinct elements and for which the boundary of the complement of T's interior ( relative to X ) consists of these same three elements.

Questions about a hyperplane arrangement A generally concern geometrical, topological, or other properties of the complement, M ( A ), which is the set that remains when the hyperplanes are removed from the whole space.

* In topological algebra — the study of topological spaces with algebraic structure, a topological Boolean algebra is a Boolean algebra endowed with a topological structure in which the operations of complement, join, and meet are continuous functions.

Heydrich, in opening the Conference, followed the reasoning and even the phraseology of the order issued earlier by Goering which authorized the Final Solution as `` a complement to '' previous `` solutions '' for eliminating the Jews from German living space through violence, economic strangulation, forced emigration, and evacuation.

Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement.

* Corona set of a space, the complement of its image in the Stone – Čech compactification.

Then the Alexandroff extension of X is a certain compact space X * together with an open embedding c: X → X * such that the complement of X in X * consists of a single point, typically denoted ∞.

By the above reasoning, the null space of A is the orthogonal complement to the row space.

The left null space of A is the orthogonal complement to the column space of A, and is the cokernel of the associated linear transformation.

The left null space is the orthogonal complement to the column space of A.

It follows that the null space of is the orthogonal complement to the column space of A.

In the case of a real affine space, in other words when the coordinates are real numbers, this affine space separates the space into two half-spaces, which are the connected components of the complement of the hyperplane, and are given by the inequalities

In four-dimensional Euclidean space, the orthogonal complement of a line is a hyperplane and vice versa, and that of a plane is a plane.

In early 2008, a multi-media Woodstock " interpretive " museum opened near the old Yasgur's Farm to complement the concert space, which hosted the New York Philharmonic, Wynton Marsalis, Diane Reeves, Chris Botti, the Goo Goo Dolls, the Boston Pops Orchestra, and Crosby, Stills, Nash & Young in its inaugural season.

He sought subjects that would complement his growing concern with form, space, and color.

If a normal operator on a finite-dimensional real or complex Hilbert space ( inner product space ) stabilizes a subspace, then it also stabilizes its orthogonal complement.

* In Star Wars: A New Hope, the Galactic Empire fields the Death Star, a moon-sized space station equipped with a superlaser capable of destroying planets, in addition to a considerable complement of warships and troops.

The redevelopment has restored Greenway's historic castellated building, removing newer additions to discreetly complement, enhance and enlarge the public green space of the Royal Botanic Gardens.