Entrelacs brunnien

In knot theory, a branch of topology, a Brunnian link is a nontrivial link that becomes a set of trivial unlinked circles if any one component is removed. In other words, cutting any loop frees all the other loops (so that no two loops can be directly linked). The name Brunnian is after Hermann Brunn. Brunn's 1892 article Über Verkettung included examples of such links. The best-known and simplest possible Brunnian link is the Borromean rings, a link of three unknots. However for every number three or above, there are an infinite number of links with the Brunnian property containing that number of loops. Here are some relatively simple three-component Brunnian links which are not the same as the Borromean rings: Image:Brunnian-3-not-Borromean.svg|12-crossing link. Image:Three-triang-18crossings-Brunnian.svg|18-crossing link. Image:Three-interlaced-squares-Brunnian-24crossings.svg|24-crossing link. The simplest Brunnian link other than the 6-crossing Borromean rings is presumably the 10-crossing L10a140 link. An example of an n-component Brunnian link is given by the "rubberband" Brunnian Links, where each component is looped around the next as aba−1b−1, with the last looping around the first, forming a circle. In 2020, new and much more complicated Brunnian links have been discovered in using highly flexible geometric-topology methods, far more than having been previously constructed. See Section 6. It is impossible for a Brunnian link to be constructed from geometric circles. Somewhat more generally, if a link has the property that each component is a circle and no two components are linked, then it is trivial. The proof, by Michael Freedman and Richard Skora, embeds the three-dimensional space containing the link as the boundary of a Poincaré ball model of four-dimensional hyperbolic space, and considers the hyperbolic convex hulls of the circles. These are two-dimensional subspaces of the hyperbolic space, and their intersection patterns reflect the pairwise linking of the circles: if two circles are linked, then their hulls have a point of intersection, but with the assumption that pairs of circles are unlinked, the hulls are disjoint.

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Bartosz Maria Walczak, Michal Lason

For positive integers w and k, two vectors A and B from Z(w) are called k-crossing if there are two coordinates i and j such that A[i] - B[i] >= k and B[j] - A[j] >= k. What is the maximum size of a family of pairwise 1-crossing and pairwise non-k-crossing ...

Academic Press Inc Elsevier Science2014

Finding Loop Invariants for Programs over Arrays Using a Theorem Prover

Laura Ildiko Kovacs

We present a new method for automatic generation of loop invariants for programs containing arrays. Unlike all previously known methods, our method allows one to generate first-order invariants containing alternations of quantifiers. The method is based on ...

Springer-Verlag New York, Ms Ingrid Cunningham, 175 Fifth Ave, New York, Ny 10010 Usa2009

An insensitivity property for light-tail shot noise traffic overflow asymptotics

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In this short note, we derive the large deviations estimate of the tail of the buffer occupancy distribution in a communications link with a very general integrated Poisson shot noise model for the total input. The result is obtained by a straightforward a ...

2000

Entrelacs de Hopf

En mathématiques, l'entrelacs de Hopf est un des modèles les plus simples étudiés en théorie des nœuds. C'est l'entrelacs non trivial et non connexe le plus simple. Il porte le nom du mathématicien Heinz Hopf. L'entrelacs de Hopf est formé par deux cercles ayant un nombre d'enlacement de plus ou moins 1. On l'obtient par exemple en considérant deux cercles situés dans des plans orthogonaux, chacun passant par le centre de l'autre. Dans la fibration de Hopf, deux fibres distinctes forment un entrelacs de Hopf dans la sphère .

Link group

In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Ph.D. thesis, . Notably, the link group is not in general the fundamental group of the link complement. The link group of an n-component link is essentially the set of (n + 1)-component links extending this link, up to link homotopy. In other words, each component of the extended link is allowed to move through regular homotopy (homotopy through immersions), knotting or unknotting itself, but is not allowed to move through other components.

Entrelacs (théorie des nœuds)

En théorie des nœuds, un entrelacs est un enchevêtrement de plusieurs nœuds. L'étude des entrelacs et des nœuds est liée, plusieurs invariants s'interprétant plus naturellement dans le cadre général des entrelacs, au moyen notamment des relations d'écheveau. Un entrelacs est la donnée d'un plongement injectif d'une ou plusieurs copies du cercle S dans R ou dans S, appelées ses composantes, ou ses boucles. Deux entrelacs sont considérés équivalents lorsqu'ils sont identiques à isotopie près.

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