Our feet and spines will unknot, and high heels will fade from consciousness along with foot-binding and rib removal to shrink your waist.
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Two simple diagrams of the unknot In the mathematical theory of knots, the unknot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it. To a knot theorist, an unknot is any embedded topological circle in the 3-sphere that is ambient isotopic (that is, deformable) to a geometrically round circle, the standard unknot. The unknot is the only knot that is the boundary of an embedded disk, which gives the characterization that only unknots have Seifert genus 0.
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Deciding if a particular knot is the unknot was a major driving force behind knot invariants, since it was thought this approach would possibly give an efficient algorithm to recognize the unknot from some presentation such as a knot diagram. Unknot recognition is known to be in both NP and co-NP. It is known that knot Floer homology and Khovanov homology detect the unknot, but these are not known to be efficiently computable for this purpose. It is not known whether the Jones polynomial or finite type invariants can detect the unknot.
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The simplest such invariant is tricolorability: the trefoil is tricolorable, but the unknot is not. In addition, virtually every major knot polynomial distinguishes the trefoil from an unknot, as do most other strong knot invariants.
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Two simple diagrams of the unknot A tricky unknot diagram by Morwen Thistlethwaite In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram. There are several types of unknotting algorithms. A major unresolved challenge is to determine if the problem admits a polynomial time algorithm; that is, whether the problem lies in the complexity class P.
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Similarly, the unknot is the identity element with respect to the knot sum operation.
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The Alexander-Conway polynomial and Jones polynomial of the unknot are trivial: :\Delta(t) = 1,\quad abla(z) = 1,\quad V(q) = 1. No other knot with 10 or fewer crossings has trivial Alexander polynomial, but the Kinoshita–Terasaka knot and Conway knot (both of which have 11 crossings) have the same Alexander and Conway polynomials as the unknot. It is an open problem whether any non-trivial knot has the same Jones polynomial as the unknot. The unknot is the only knot whose knot group is an infinite cyclic group, and its knot complement is homeomorphic to a solid torus.
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Although the Khovanov homology detects the unknot, it is not yet known if the Jones polynomial does.
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A knot can be untied if the loop is broken. The simplest knot, called the unknot or trivial knot, is a round circle embedded in . In the ordinary sense of the word, the unknot is not "knotted" at all. The simplest nontrivial knots are the trefoil knot ( in the table), the figure-eight knot () and the cinquefoil knot ().
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The four half-twist stevedore knot is created by passing the one end of an unknot with four half-twists through the other. All twist knots have unknotting number one, since the knot can be untied by unlinking the two ends. Every twist knot is also a 2-bridge knot. Of the twist knots, only the unknot and the stevedore knot are slice knots.
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The trefoil knot is nontrivial, meaning that it is not possible to "untie" a trefoil knot in three dimensions without cutting it. Mathematically, this means that a trefoil knot is not isotopic to the unknot. In particular, there is no sequence of Reidemeister moves that will untie a trefoil. Proving this requires the construction of a knot invariant that distinguishes the trefoil from the unknot.
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In mathematics, a stuck unknot is a closed polygonal chain that is topologically equal to the unknot but cannot be deformed to a simple polygon by rigid motions of the segments.G. Aloupis, G. Ewald, and G. T. Toussaint, "More classes of stuck unknotted hexagons," Contributions to Algebra and Geometry, Vol. 45, No. 2, 2004, pp. 429–434.G. T. Toussaint, "A new class of stuck unknots in Pol-6," Contributions to Algebra and Geometry, Vol.
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A twist knot with n half-twists has crossing number n+2. All twist knots are invertible, but the only amphichiral twist knots are the unknot and the figure-eight knot.
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The (−3, 0, −3) pretzel knot (square knot (mathematics)) is the connected sum of two trefoil knots. The (0, q, 0) pretzel link is the split union of an unknot and another knot.
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Thistlethwaite unknot Morwen B. Thistlethwaite is a knot theorist and professor of mathematics for the University of Tennessee in Knoxville. He has made important contributions to both knot theory and Rubik's Cube group theory.
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A knot is called achiral if it can be continuously deformed into its mirror image, otherwise it is called chiral. For example, the unknot and the figure-eight knot are achiral, whereas the trefoil knot is chiral.
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A knot is called achiral if it can be continuously deformed into its mirror image, otherwise it is called a chiral knot. For example, the unknot and the figure-eight knot are achiral, whereas the trefoil knot is chiral.
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75–78 Now we can define the Arf invariant of a knot to be 0 if it is pass-equivalent to the unknot, or 1 if it is pass-equivalent to the trefoil. This definition is equivalent to the one above.
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Hass is known for proving the equal-volume special case of the double bubble conjecture, for proving that the unknotting problem is in NP, and for giving an exponential bound on the number of Reidemeister moves needed to reduce the unknot to a circle.
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The Dacre knot, a type of decorative unknot, is a heraldic knot used primarily in English heraldry. It is most notable for its appearance on the Dacre family heraldic badge, where its two lower dexter loops entwine a scallop, and its two lower sinister loops entwine a log.
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Peripheral loops live in γ union the tube. The peripheral subgroups for a tame knot K in R3 are isomorphic to Z ⊕ Z if the knot is nontrivial, Z if it is the unknot. They are generated by two elements, called a longitude [l] and a meridian [m]. (If K is the unknot, then [l] is a power of [m], and a peripheral subgroup is generated by [m] alone.) A longitude is a loop that runs from the basepoint x along a path γ to a point y on the boundary of a tubular neighborhood of K, then follows along the tube, making one full lap to return to y, then returns to x via γ.
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In the mathematical field of knot theory, the crosscap number of a knot K is the minimum of :1 - \chi(S), \, taken over all compact, connected, non- orientable surfaces S bounding K; here \chi is the Euler characteristic. The crosscap number of the unknot is zero, as the Euler characteristic of the disk is one.
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Figure-eight knot of practical knot-tying, with ends joined In knot theory, a figure-eight knot (also called Listing's knot) is the unique knot with a crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the unknot and the trefoil knot. The figure- eight knot is a prime knot.
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By way of example, the unknot has crossing number zero, the trefoil knot three and the figure-eight knot four. There are no other knots with a crossing number this low, and just two knots have crossing number five, but the number of knots with a particular crossing number increases rapidly as the crossing number increases.
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Analogies for the plane would be lifting a string up off the surface, or removing a dot from inside a circle. In fact, in four dimensions, any non-intersecting closed loop of one- dimensional string is equivalent to an unknot. First "push" the loop into a three-dimensional subspace, which is always possible, though technical to explain.
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There are several equivalent statements of the Smale conjecture. One is that the component of the unknot in the space of smooth embeddings of the circle in 3-space has the homotopy-type of the round circles, equivalently, O(3). Another equivalent statement is that the group of diffeomorphisms of the 3-ball which restrict to the identity on the boundary is contractible.
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The first application of Khovanov homology was provided by Jacob Rasmussen, who defined the s-invariant using Khovanov homology. This integer valued invariant of a knot gives a bound on the slice genus, and is sufficient to prove the Milnor conjecture. In 2010, Kronheimer and Mrowka proved that the Khovanov homology detects the unknot. The categorified theory has more information than the non-categorified theory.
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A twist knot with six half-twists. In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together. (That is, a twist knot is any Whitehead double of an unknot.) The twist knots are an infinite family of knots, and are considered the simplest type of knots after the torus knots.
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There is a prime decomposition for knots, analogous to prime and composite numbers . For oriented knots, this decomposition is also unique. Higher-dimensional knots can also be added but there are some differences. While you cannot form the unknot in three dimensions by adding two non-trivial knots, you can in higher dimensions, at least when one considers smooth knots in codimension at least 3.
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They developed an instanton Floer invariant for knots which was used in their proof that Khovanov homology detects the unknot. Kronheimer attended the City of London School. He completed his PhD at Oxford University under the direction of Michael Atiyah. He has had a long association with Merton College, the oldest of the constituent colleges of Oxford University, being an undergraduate, graduate, and full fellow of the college.
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Sequoia awarded an honorary Doctor of Science in Engineering to David B. Steinman received on 15 April 1952. In May 1956, Sequoia University awarded actor Mickey Rooney an honorary PhD in Fine Arts. In 1968, literary scholar Devendra Varma received a fellowship of the Sequoia Research Institute, a subsidiary of Sequoia University. American zen poet Paul Reps published his second book "Unknot The World In You" through Sequoia University Press.
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The link is created with two projections of the unknot: one circular loop and one figure eight-shaped loop (i.e., a loop with a Reidemeister Type I move applied) intertwined such that they are inseparable and neither loses its form. Excluding the instance where the figure eight thread intersects itself, the Whitehead link has four crossings. Because each under-hand crossing has a paired upper-hand crossing, its linking number is 0.
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The Shakespeare knot, a type of decorative unknot, is a heraldic knot. It is a derivative of the Bowen knot and closely akin to the Dacre knot. The knot is most notable for its appearance on the Shakespeare badge. A signet ring preserved in Stratford-upon-Avon said to have belonged to William Shakespeare bears the knot, where its lower dexter and sinister loops entwine a W (for William) and an S (for Shakespeare).
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The Burau representation for has been known to be faithful for some time. The faithfulness of the Burau representation when is an open problem. The Burau representation appears as a summand of the Jones representation, and for , the faithfulness of the Burau representation is equivalent to that of the Jones representation, which on the other hand is related to the question of whether or not the Jones polynomial is an unknot detector.
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This approach to the Arf invariant is due to Louis Kauffman. We define two knots to be pass equivalent if they are related by a finite sequence of pass-moves,Kauffman (1987) p.74 which are illustrated below: (no figure right now) Every knot is pass-equivalent to either the unknot or the trefoil; these two knots are not pass-equivalent and additionally, the right- and left-handed trefoils are pass-equivalent.Kauffman (1987) pp.
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To remove knots from highly crowded chromatin, one would need an active process that should not only provide the energy to move the system from the state of topological equilibrium but also guide topoisomerase-mediated passages in such a way that knots would be efficiently unknotted instead of making the knots even more complex. It has been shown that the process of chromatin-loop extrusion is ideally suited to actively unknot chromatin fibres in interphase chromosomes.
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Topology of beta-strands in "Greek-key" protein motif. Protein topology is a property of protein molecule that does not change under deformation (without cutting or breaking a bond). Two main topology frameworks have been developed and applied to protein molecules: 1) Knot theory which categorises chain entanglements 2) Circuit topology which categorises intra-chain contacts based on their arrangements. The usage of knot theory is however limited to a small percentage of proteins as most of them are unknot.
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Every non-trivial knot has bridge number at least two, so the knots that minimize the bridge number (other than the unknot) are the 2-bridge knots. It can be shown that every n-bridge knot can be decomposed into two trivial n-tangles and hence 2-bridge knots are rational knots. If K is the connected sum of K1 and K2, then the bridge number of K is one less than the sum of the bridge numbers of K1 and K2..
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Therefore knot groups have some usage in knot theory to distinguish between knots: if \pi_1(\R^3 \setminus K) is not isomorphic to some other knot group \pi_1(\R^3 \setminus K') of another knot K', then K can not be transformed into K'. Thus the trefoil knot can not be continuously transformed into the circle (also known as the unknot), since the latter has knot group \Z. There are, however, knots that can not be deformed into each other, but have isomorphic knot groups.
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The Hinckaert knot, a type of decorative unknot, is a heraldic knot used primarily in Dutch heraldry. It is most notable for its appearance on the Hinckaert family heraldic badge, where a semi-angular form is used as canting arms, a common practice with heraldic badges. The name "Hinckaert" is delineated as a derivation of hincken, "to limp", in the badge. Hence the center crutch, and the buckle on the knot, implying that it is a strap used to attach the crutch to the leg.
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Another spectral sequence (Ozsváth-Szabó 2005) relates a variant of Khovanov homology with the Heegaard Floer homology of the branched double cover along a knot. A third (Bloom 2009) converges to a variant of the monopole Floer homology of the branched double cover. In 2010 Kronheimer and Mrowka exhibited a spectral sequence abutting to their instanton knot Floer homology group and used this to show that Khovanov Homology (like the instanton knot Floer homology) detects the unknot. Khovanov homology is related to the representation theory of the Lie algebra sl2.
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It is related by mutation to the Kinoshita–Terasaka knot, with which it shares the same Jones polynomial. Both knots also have the curious property of having the same Alexander polynomial and Conway polynomial as the unknot. The issue of the sliceness of the Conway knot was resolved in 2020 by Lisa Piccirillo, 50 years after John Horton Conway first proposed the knot. Her proof made use of Rasmussen's s-invariant, and showed that the knot is not a smoothly slice knot, though it is topologically slice (the Kinoshita–Terasaka knot is both).
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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.. 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, ... Modern automated methods can now enumerate billions of knots in a matter of days.
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Since V is an unknotted solid torus, S^3 \setminus V is a tubular neighbourhood of an unknot J. The 2-component link K' \cup J together with the embedding f is called the pattern associated to the satellite operation. A convention: people usually demand that the embedding f \colon V \to S^3 is untwisted in the sense that f must send the standard longitude of V to the standard longitude of f(V). Said another way, given any two disjoint curves c_1,c_2 \subset V, f preserves their linking numbers i.e.: lk(f(c_1),f(c_2))=lk(c_1,c_2).
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Lackenby's research contributions include a proof of a strengthened version of the 2 theorem on sufficient conditions for Dehn surgery to produce a hyperbolic manifold, a bound on the hyperbolic volume of a knot complement of an alternating knot, and a proof that every diagram of the unknot can be transformed into a diagram without crossings by only a polynomial number of Reidemeister moves. Lackenby won the Whitehead Prize of the London Mathematical Society in 2003. In 2006, he won the Philip Leverhulme Prize in mathematics and statistics. He was an invited speaker at the International Congress of Mathematicians in 2010.
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The German paratrooper knife, which featured a marlinspike in addition to the cutting blade, was used to cut rigging and unknot lines, though it could be employed as a weapon in an emergency. The U.S. Army in 1940 tasked the Geo. Schrade Knife Co. to produce a small single-edge switchblade for U.S. airborne troops, to be used similarly to the Fallschirmjäger-Messer. The knife was not intended primarily as a fighting knife, but rather as a utility tool, to enable a paratrooper to rapidly cut himself out of his lines and harness in the event he could not escape them after landing.
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Mikhail Khovanov and Lev Rozansky have since defined several other related cohomology theories whose Euler characteristics recover other classical invariants. Catharina Stroppel gave a representation theoretic interpretation of Khovanov homology by categorifying quantum group invariants. There is also growing interest from both knot theorists and scientists in understanding "physical" or geometric properties of knots and relating it to topological invariants and knot type. An old result in this direction is the Fary–Milnor theorem states that if the total curvature of a knot in \R^3 satisfies :\oint_K \kappa \,ds \leq 4\pi, where is the curvature at , then is an unknot.
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Knots can be tied in the core of such a vortex, leading to the hypothesis that each chemical element corresponds to a different kind of knot. The simple toroidal vortex, represented by the circular "unknot" 01, was thought to represent hydrogen. Many elements had yet to be discovered, so the next knot, the trefoil knot 31, was thought to represent carbon. However, as more elements were discovered and the periodicity of their characteristics established in the periodic table of the elements, it became clear that this could not be explained by any rational classification of knots.
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As a closed loop, a mathematical knot has no proper ends, and cannot be undone or untied; however, any physical knot in a piece of string can be thought of as a mathematical knot by fusing the two ends. A configuration of several knots winding around each other is called a link. Various mathematical techniques are used to classify and distinguish knots and links. For instance, the Alexander polynomial associates certain numbers with any given knot; these numbers are different for the trefoil knot, the figure-eight knot, and the unknot (a simple loop), showing that one cannot be moved into the other (without strands passing through each other).
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Godfried Toussaint Godfried Theodore Patrick Toussaint (1944 – July 2019) was a Canadian Computer Scientist, a Professor of Computer Science, and the Head of the Computer Science Program at New York University Abu Dhabi (NYUAD)New York University Abu Dhabi in Abu Dhabi, United Arab Emirates. He is considered to be the father of computational geometry in Canada. He did research on various aspects of computational geometry, discrete geometry, and their applications: pattern recognition (k-nearest neighbor algorithm, cluster analysis), motion planning, visualization (computer graphics), knot theory (stuck unknot problem), linkage (mechanical) reconfiguration, the art gallery problem, polygon triangulation, the largest empty circle problem, unimodality (unimodal function), and others. Other interests included meander (art), compass and straightedge constructions, instance-based learning, music information retrieval, and computational music theory.
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Other examples are knot polynomials, such as the Jones polynomial, which are currently among the most useful invariants for distinguishing knots from one another, though currently it is not known whether there exists a knot polynomial which distinguishes all knots from each other. However, there are invariants which distinguish the unknot from all other knots, such as Khovanov homology and knot Floer homology. Other invariants can be defined by considering some integer-valued function of knot diagrams and taking its minimum value over all possible diagrams of a given knot. This category includes the crossing number, which is the minimum number of crossings for any diagram of the knot, and the bridge number, which is the minimum number of bridges for any diagram of the knot.
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Bourchier knot compilation, Tawstock Church, Devon A heraldic knot (referred to in heraldry as simply a knot) is a knot, unknot, or design incorporating a knot used in European heraldry. While a given knot can be used on more than one family's achievement of arms, the family on whose coat the knot originated usually gives its name to the said knot (the exception being the Tristram knot). These knots can be used to charge shields and crests, but can also be used in badges or as standalone symbols of the families for whom they are named (like Scottish plaids). The simplest of these patterns, the Bowen knot, is often referred to as the heraldic knot in symbolism and art outside of heraldry.
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In the work of Lenhard Ng, relative SFT is used to obtain invariants of smooth knots: a knot or link inside a topological three-manifold gives rise to a Legendrian torus inside a contact five-manifold, consisisting of the unit conormal bundle to the knot inside the unit cotangent bundle of the ambient three-manifold. The relative SFT of this pair is a differential graded algebra; Ng derives a powerful knot invariant from a combinatorial version of the zero-th degree part of the homology. It has the form of a finitely presented tensor algebra over a certain ring of multivariable Laurent polynomials with integer coefficients. This invariant assigns distinct invariants to (at least) knots of at most ten crossings, and dominates the Alexander polynomial and the A-polynomial (and thus distinguishes the unknot).
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Like his other publicly available books of teachings, Chögyal Namkhai Norbu Rinpoche's Dream Yoga and the Practice of Natural Light is a generous transformative reading experience, a resource that will unknot the thought of readers ready to engage it. Additionally (as seems to be the rule with such public works of his), for those who have received Rinpoche's living transmission, the volume can unlock the real essence of such transmission, reconciling the puzzle pieces of our day with the complete picture we embody from a larger perspective." Excerpt from Jesse Abbott's review reprint from The Mirror In addition to contact with Tibetan Lama Chogyal Namkhai Norbu in the dream state, Katz cites personal lucid dream experiences such as the following as having inspired the original book. > "In my dream I was standing on a beach near the shoreline.
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Brunnian links were classified up to link-homotopy by John Milnor in , and the invariants he introduced are now called Milnor invariants. An (n + 1)-component Brunnian link can be thought of as an element of the link group – which in this case (but not in general) is the fundamental group of the link complement – of the n-component unlink, since by Brunnianness removing the last link unlinks the others. The link group of the n-component unlink is the free group on n generators, Fn, as the link group of a single link is the knot group of the unknot, which is the integers, and the link group of an unlinked union is the free product of the link groups of the components. Not every element of the link group gives a Brunnian link, as removing any other component must also unlink the remaining n elements.
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Roughly, the Whitney trick allows one to "unknot" knotted spheres – more precisely, remove self-intersections of immersions; it does this via a homotopy of a disk – the disk has 2 dimensions, and the homotopy adds 1 more – and thus in codimension greater than 2, this can be done without intersecting itself; hence embeddings in codimension greater than 2 can be understood by surgery. In surgery theory, the key step is in the middle dimension, and thus when the middle dimension has codimension more than 2 (loosely, 2½ is enough, hence total dimension 5 is enough), the Whitney trick works. The key consequence of this is Smale's h-cobordism theorem, which works in dimension 5 and above, and forms the basis for surgery theory. A modification of the Whitney trick can work in 4 dimensions, and is called Casson handles – because there are not enough dimensions, a Whitney disk introduces new kinks, which can be resolved by another Whitney disk, leading to a sequence ("tower") of disks.
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In the passage, Ezra argues with Uriel about matters of justice in a way that he never could with God; however, the angel argues back with a series of riddles that eventually show Ezra the misguidedness of his thinking (4 Ezra 3:1-4:21). Importantly, Uriel does not simply transmit information or “speak at” Ezra; the two are engaged in an animated dialogue that reflects that of a teacher and a student, with the former guiding the latter to a realization. Ezra could never argue with God the way he argues with Uriel; however, this argument and its accompanying emotional catharsis is partially what leads him to discover the truth and main message of the passage on his own. In Daniel, angels also assume the roles of interpreters and teachers, notably in their abilities to explain visions concerning the eschaton, and help human prophets unknot knowledge from it. In Daniel, it is the archangel Gabriel who is sent down from heaven by God to explain Daniel's perplexing visions and help relieve some of his distress (Daniel 8:16-17).
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