Piecewise-linear embeddings of knots and links with rotoinversion symmetry

Michael O'Keeffe; Michael M. J. Treacy

doi:10.1107/s2053273321006136

Isogonal Weavings on the Sphere: Knots, Links, Polycatenanes

10.26434/chemrxiv.12616268.v1 ◽

2020 ◽

Author(s):

Michael O'Keeffe ◽

Michael Treacy

Keyword(s):

Piecewise Linear ◽

Infinite Family ◽

Torus Knots ◽

Vertex Transitive ◽

Knots And Links ◽

Complete Account

We describe mathematical knots and links as piecewise linear – straight, non-intersecting sticks meeting at corners. Isogonal structures have all corners related by symmetry ("vertex" transitive). Corner- and stick-transitive structures are termed regular. We find no regular knots. Regular links are cubic or icosahedral and a complete account of these is given, including optimal (thickest-stick) embeddings. Stick 2-transitive isogonal structures are again cubic and icosahedral and also encompass the infinite family of torus knots and links. The major types of these structures are identified and reported with optimal embeddings. We note the relevance of this work to materials- and bio-chemistry.

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Isogonal weavings on the sphere: knots, links, polycatenanes

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273320010669 ◽

2020 ◽

Vol 76 (5) ◽

pp. 611-621

Author(s):

Michael O'Keeffe ◽

Michael M. J. Treacy

Keyword(s):

Piecewise Linear ◽

Infinite Family ◽

Materials Chemistry ◽

Torus Knots ◽

Vertex Transitive ◽

Knots And Links ◽

Complete Account

Mathematical knots and links are described as piecewise linear – straight, non-intersecting sticks meeting at corners. Isogonal structures have all corners related by symmetry (`vertex'-transitive). Corner- and stick-transitive structures are termed regular. No regular knots are found. Regular links are cubic or icosahedral and a complete account of these (36 in number) is given, including optimal (thickest-stick) embeddings. Stick 2-transitive isogonal structures are again cubic and icosahedral and also encompass the infinite family of torus knots and links. The major types of these structures are identified and reported with optimal embeddings. The relevance of this work to materials chemistry and biochemistry is noted.

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Isogonal Weavings on the Sphere: Knots, Links, Polycatenanes

10.26434/chemrxiv.12616268 ◽

2020 ◽

Author(s):

Michael O'Keeffe ◽

Michael Treacy

Keyword(s):

Piecewise Linear ◽

Infinite Family ◽

Torus Knots ◽

Vertex Transitive ◽

Knots And Links ◽

Complete Account

We describe mathematical knots and links as piecewise linear – straight, non-intersecting sticks meeting at corners. Isogonal structures have all corners related by symmetry ("vertex" transitive). Corner- and stick-transitive structures are termed regular. We find no regular knots. Regular links are cubic or icosahedral and a complete account of these is given, including optimal (thickest-stick) embeddings. Stick 2-transitive isogonal structures are again cubic and icosahedral and also encompass the infinite family of torus knots and links. The major types of these structures are identified and reported with optimal embeddings. We note the relevance of this work to materials- and bio-chemistry.

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Minimal Plat Representations of Prime Knots and Links are not Unique

Canadian Journal of Mathematics ◽

10.4153/cjm-1976-020-5 ◽

1976 ◽

Vol 28 (1) ◽

pp. 161-167 ◽

Cited By ~ 6

Author(s):

José M. Montesinos

Keyword(s):

Infinite Family ◽

Equivalence Classes ◽

Covering Space ◽

Heegaard Splittings ◽

Cyclic Covering ◽

Knots And Links

Letdenote the 2-fold cyclic covering space branched over a linkLin S3. We wish to describe an infinite family of prime knots and links in which each memberLexhibits two minimal 6-plat representations, where the associated Heegaard splittings ofare minimal and inequivalent. Thus each knot or link of that family admits at least two equivalence classes of 6-plat representations which are minimal.

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On Discontinuous Piecewise Linear Models for Memristor Oscillators

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127417300221 ◽

2017 ◽

Vol 27 (06) ◽

pp. 1730022 ◽

Cited By ~ 7

Author(s):

Andrés Amador ◽

Emilio Freire ◽

Enrique Ponce ◽

Javier Ros

Keyword(s):

Periodic Orbits ◽

Invariant Manifolds ◽

Linear Models ◽

Initial Conditions ◽

Piecewise Linear ◽

Three Dimensional ◽

Infinite Family ◽

Closed Surfaces ◽

The Rich ◽

Piecewise Linear Models

In this paper, we provide for the first time rigorous mathematical results regarding the rich dynamics of piecewise linear memristor oscillators. In particular, for each nonlinear oscillator given in [Itoh & Chua, 2008], we show the existence of an infinite family of invariant manifolds and that the dynamics on such manifolds can be modeled without resorting to discontinuous models. Our approach provides topologically equivalent continuous models with one dimension less but with one extra parameter associated to the initial conditions. It is possible to justify the periodic behavior exhibited by three-dimensional memristor oscillators, by taking advantage of known results for planar continuous piecewise linear systems. The analysis developed not only confirms the numerical results contained in previous works [Messias et al., 2010; Scarabello & Messias, 2014] but also goes much further by showing the existence of closed surfaces in the state space which are foliated by periodic orbits. The important role of initial conditions that justify the infinite number of periodic orbits exhibited by these models, is stressed. The possibility of unsuspected bistable regimes under specific configurations of parameters is also emphasized.

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