An enumeration of knots and links, and some of their algebraic properties

Conway, James

doi:10.1016/b978-0-08-012975-4.50034-5

Cited by 578 publications

(687 citation statements)

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“…In this section we apply our results to certain families of knots defined in terms of the notation of Conway [4]; since that notation naturally encodes the characteristic toric orbifold decomposition of a knot it is eminently suited to our techniques. In particular, we consider all the knots up to 11 crossings in Conway's tables [4] with the property that their description in the tables makes it clear that they contain an essential Conway sphere.…”

Section: Examplesmentioning

confidence: 99%

“…We apply our results to the knots that are algebraic in the sense of Conway [4] (see also Thistlethwaite [33]), and which have an essential Conway sphere. We call such a knot K a large algebraic knot.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 10 we apply Theorem 8.2 to the knots in Conway's tables [4] of knots up to 11 crossings that can be immediately seen to be large algebraic by virtue of their description in terms of Conway's notation. There are 174 such knots, and we show that exactly 24 of them have unknotting number 1.…”

Section: Introductionmentioning

confidence: 99%

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Knots with unknotting number 1 and essential Conway spheres

Gordon

Luecke

2006

Algebr. Geom. Topol.

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First we clarify the extension of the main result of [11] to knots in solid tori that is described in the Appendix of [11]. In Section 3, we define a family of hyperbolic knots J " .`; m/ in a solid torus, " 2 f1; 2g and`; m integers, each of which admits a half-integer surgery yielding a toroidal manifold. (The knots J " .`; m/ in the solid torus are the analogs of the knots k.`; m; n; p/ in the 3-sphere.) We then use [11] to show that these are the only such:Theorem 4.2 Let J be a knot in a solid torus whose exterior is irreducible and atoroidal. Let be the meridian of J and suppose that J. / contains an essential torus for some with . ; / 2. Then . ; / D 2 and J D J " .`; m/ for some ";`; m.This theorem along with the main result of [11] then allows us to describe in Theorem 5.2 the relationship between the torus decomposition of the exterior of a knot K in S 3 and the torus decompositon of any non-integral surgery on K . In particular, Theorem 5.2 says that the canonical tori of the exterior of K and of the Dehn surgery will be the same unless K is a cable knot (in which case an essential torus of the knot exterior can become compressible), or K is a k.`; m; n; p/, or K is a satellite with pattern J " .`; m/ (in the latter cases a new essential torus is created).We apply these theorems about non-integral Dehn surgeries to address questions about unknotting number. The knots k.`;m;n;p/ (J " .`; m/) are strongly invertible. Their quotients under the involutions give rise to EM-knots, K.`;m;n;p/ (EM-tangles, A " .`; m/ resp.) which have essential Conway spheres and yet can be unknotted (trivialized, resp.) by a single crossing change. Theorem 6.2 descibes when a knot with an essential Conway sphere or 2-torus can have unknotting number 1. This is naturally stated in the context of the characteristic decomposition of a knot along

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Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Knots with unknotting number 1 and essential Conway spheres

Gordon

Luecke

2006

Algebr. Geom. Topol.

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“…These transfer matrices are not actually matrices but rather diagrammatic objects (tangles [38,39]) defined in the corresponding planar TL algebra [2,3]. Matrix representations are obtained by acting with these tangles on suitable vector spaces of link states [1,35].…”

Section: Lm(pmentioning

confidence: 99%

“…The elements of the planar TL algebra are called tangles [38,39] and are diagrammatic objects formed by adding or linking together a number of elementary face operators. Noting that 5) we stress that individual connectivity diagrams are themselves tangles.…”

Section: Temperley-lieb Loop Modelsmentioning

confidence: 99%

Fusion hierarchies,T-systems, andY-systems of logarithmic minimal models

2014

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A Temperley-Lieb (TL) loop model is a Yang-Baxter integrable lattice model with nonlocal degrees of freedom. On a strip of width N ∈ N, the evolution operator is the double-row transfer tangle D(u), an element of the TL algebra TL N (β) with loop fugacity β = 2 cos λ, λ ∈ R. Similarly on a cylinder, the single-row transfer tangle T (u) is an element of the so-called enlarged periodic TL algebra. The logarithmic minimal models LM(p, p ′ ) comprise a subfamily of the TL loop models for which the crossing parameter λ = (p ′ − p)π/p ′ is a rational multiple of π parameterised by coprime integers 1 ≤ p < p ′ . For these special values, additional symmetries allow for particular degeneracies in the spectra that account for the logarithmic nature of these theories. For critical dense polymers LM(1, 2), with β = 0, D(u) and T (u) satisfy inversion identities that have led to the exact determination of the eigenvalues in any representation and for arbitrary finite system size N . The generalisation for p ′ > 2 takes the form of functional relations for D(u) and T (u) of polynomial degree p ′ . These derive from fusion hierarchies of commuting transfer tangles D m,n (u) and T m,n (u) where D(u) = D 1,1 (u) and T (u) = T 1,1 (u). The fused transfer tangles are constructed from (m, n)-fused face operators involving Wenzl-Jones projectors P k on k = m or k = n nodes. Some projectors P k are singular for k ≥ p ′ , but we argue that D m,n (u) and T m,n (u) are nonsingular for every m, n ∈ N in certain cabled link state representations. For generic λ, we derive the fusion hierarchies and the associated T -and Y -systems. For the logarithmic theories, the closure of the fusion hierarchies at n = p ′ translates into functional relations of polynomial degree p ′ for D m,1 (u) and T m,1 (u). We also derive the closure of the Y -systems for the logarithmic theories. The T -and Y -systems are the key to exact integrability and we observe that the underlying structure of these functional equations relate to Dynkin diagrams of affine Lie algebras.

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Molekulare Knoten

Fielden¹,

Leigh²,

Woltering³

2017

Angewandte Chemie

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Der erste synthetische Kleeblattknoten wurde in den späten 1980er Jahren hergestellt. Kompliziertere Knotentopologien wurden jedoch erst in den letzten Jahren in molekularer Form erhalten. Die Verknotung molekularer Stränge erzeugt sterische Einschränkungen und kann so wichtige physikalische und chemische Eigenschaften verleihen, unter anderem Chiralität, starkes und selektives Binden von Ionen und katalytische Aktivität. Da die Anzahl und Komplexität molekularer Knoten steigt, wird es für Chemiker immer wichtiger, die Knotennomenklatur anderer Disziplinen zu verwenden. Hier geben wir einen Überblick über die Synthesestrategien für molekulare Knoten und beschreiben die Grundlagen der Knoten‐, Zopf‐ und Tangle‐Theorie in einem für Chemiker und molekulare Strukturen angemessenen Umfang.

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An enumeration of knots and links, and some of their algebraic properties

Cited by 578 publications

References 4 publications

Knots with unknotting number 1 and essential Conway spheres

Knots with unknotting number 1 and essential Conway spheres

Fusion hierarchies,T-systems, andY-systems of logarithmic minimal models

Molekulare Knoten

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