scholarly journals Shake slice and shake concordant links

2020 ◽  
Vol 29 (12) ◽  
pp. 2050087
Author(s):  
Anthony M. Bosman

We can construct a [Formula: see text]-manifold by attaching [Formula: see text]-handles to a [Formula: see text]-ball with framing [Formula: see text] along the components of a link in the boundary of the [Formula: see text]-ball. We define a link as [Formula: see text]-shake slice if there exists embedded spheres that represent the generators of the second homology of the [Formula: see text]-manifold. This naturally extends [Formula: see text]-shake slice, a generalization of slice that has previously only been studied for knots, to links of more than one component. We also define a relative notion of shake[Formula: see text]-concordance for links and versions with stricter conditions on the embedded spheres that we call strongly[Formula: see text]-shake slice and strongly[Formula: see text]-shake concordance. We provide infinite families of links that distinguish concordance, shake concordance, and strong shake concordance. Moreover, for [Formula: see text] we completely characterize shake slice and shake concordant links in terms of concordance and string link infection. This characterization allows us to prove that the first non-vanishing Milnor [Formula: see text] invariants are invariants of shake concordance. We also argue that shake concordance does not imply link homotopy.

1975 ◽  
Vol 42 (1) ◽  
pp. 46-48 ◽  
Author(s):  
Nathan Stemmer

1993 ◽  
Vol 02 (01) ◽  
pp. 37-63 ◽  
Author(s):  
JAMES R. HUGHES

We study link-homotopy classes of links in the three sphere using reduced groups endowed with peripheral structures derived from meridian-longitude pairs. Two types of peripheral structures are considered — Milnor’s original version (called “pre-peripheral structures” in Levine’s terminology) and Levine’s refinement (called simply “peripheral structures”). We show here that pre-peripheral structures are not strong enough to classify links up to link-homotopy, and that Levine’s peripheral structures, although strong enough to distinguish those classes not distinguished by pre-peripheral structures, are also in all likelihood not strong enough to distinguish all link-homotopy classes. Following Levine’s classification program, we compare structure-preserving and realizable automorphisms, using an obstruction-theoretic approach suggested by work of Habegger and Lin. We find that these automorphism groups are in general different, so that a more complex program for classification by structured groups is required.


2021 ◽  
Author(s):  
Deep Bhattacharjee ◽  
Sanjeevan Singha Roy

Higher dimensions are impossible to visualize as the size of dimension varies inversely proportional to its level. The more the dimension ranges, the least its size. We are a set of points living in a particular point of space and a particular frame of time. i.e, we live in space-time. The space has more dimensions that meets the human eye. We are living in a world of hyper-space. Our world being a smaller dimension is floating in higher dimensions. The quest for the visually of higher dimensions has been a fantasy to mankind but this aspect of nature is completely locked. We can transform dimensions i.e., from higher to lower dimensions, or from lower to higher dimensions, but only through mathematics. The relative notion of mathematics helps us to do the thing, which is perhaps impossible in the experimental part of physical reality. Humans being an element of 3 Dimensions – length, breath, height can only perceive one higher dimensions, that is space-time. but beyond that the notion of dimension itself changes. The dimensions got curled up in every intersection of the coordinates of space in such a way that the higher dimensions remain stable to us. But in reality it is highly unstable. In the higher dimensions, above 4, the space is tearing apart and joining again spontaneously, but the tearing portion itself covered by 2 dimensional Branes which acts as a stabilizer for the unstable dimensions. Dimensions will get smaller and smaller with the space-time interwoven in it. But at Planks length that is 10^-33 meter, the notion of space-time itself breaks down thereby making impossible for the higher dimensions to coexist along with space. Without space, there will be no identity of any dimension. The space itself is the fabric for the milestone of residing higher dimensions. Imagine our room, which is 3 dimensional. But what is there inside the room. The space and of course the time. Space-time being a totally separate entity is not quite separate when compared with other dimensions because it makes the residing place for the higher dimensions or the hyperspace itself. We all are confined within a lower dimensional world within a randomness of higher dimensions. Time being alike like space is an arrow which has the capability of slicing space into different forms. Thereby taking a snapshot of our every nano-second we vibrate within space-time. As each slice of time represents each slice of space, similarly each slice of space represents each slice of time. The nature of space-time is beyond human consciousness. It is the identity by which we breathe, we play, we survive. It is the whole localization of species that encompasses itself with space thereby making space-time a relative quantity depending upon the reference frame. The only thing that can encompass space-time or even change the relative definition of space-time is the speed, the speed far beyond the speed of light. The more the speed, the less the array of time flows. Space-time being an invisible entity makes the other dimensions visible residing in it only into the level of 3, that is l, b, h. After that there is a infamous structure formed by the curling of higher dimensions called CALABI-YAU manifold. This manifold depicts the usual nature of the dimensional quadrants of the higher order by containing a number of small spherical spheres inside it. The mathematics of string theory is still unable to solve the genus and the containing spheres of the manifold which can be the ultimate quest for the hidden dimensions. Hidden, as, the higher dimensions are hidden from human perspective of macro level but if we probe deeper into the fabric of the space-time of General Relativity then we will find the 5th dimension according to the Kaluza-Klein theory. And if we probe even deeper into it at the perspective of string theory we will be amazed to see the real nature of quantum world. They are so marvelously beautiful, they contain so many forms of higher dimensions ranging from 6 to 10. And even many more of that, but we are still not sure about it where they may exist in a ghost state. After all, the quantum nature is far more beautiful that one can even imagine with a full faze of weirdness.


2010 ◽  
Vol 10 (01) ◽  
pp. 77-95 ◽  
Author(s):  
MANUEL STADLBAUER

We introduce a relative notion of the "big images and preimages"-property for random topological Markov chains. This condition then implies that a relative version of the Ruelle–Perron–Frobenius theorem holds with respect to summable and locally Hölder continuous potentials.


2002 ◽  
Vol 11 (03) ◽  
pp. 353-362 ◽  
Author(s):  
YASUTAKA NAKANISHI ◽  
YOSHIYUKI OHYAMA

In this note, we will study Δ link homotopy (or self Δ-equivalence), which is an equivalence relation of ordered and oriented link types. Previously, a necessary condition is given in the terms of Conway polynomials for two link types to be Δ link homotopic. A pair of numerical invariants δ1 and δ2 classifies all (ordered and oriented) prime 2-component link types with seven crossings or less up to Δ link homotopy. We will show here that for any pair of integers n1 and n2 there exists a 2-component link κ such that δ1(κ) = n1 and δ2(κ) = n2 provided that at least one of n1 and n2 is even.


2017 ◽  
Vol 26 (12) ◽  
pp. 1750077
Author(s):  
Ash Lightfoot

It is an open problem whether Kirk’s [Formula: see text]-invariant is the complete obstruction to a link map [Formula: see text] being link homotopic to the trivial link. The link homotopy invariant associates to such a link map [Formula: see text] a pair [Formula: see text], and we write [Formula: see text]. With the objective of constructing counterexamples, Li proposed a link homotopy invariant [Formula: see text] such that [Formula: see text] is defined on the kernel of [Formula: see text] and which also obstructs link null-homotopy. We show that, when defined, the invariant [Formula: see text] is determined by [Formula: see text], and is strictly weaker. In particular, this implies that if a link map [Formula: see text] has [Formula: see text], then after a link homotopy the self-intersections of [Formula: see text] may be equipped with framed, immersed Whitney disks in [Formula: see text] whose interiors are disjoint from [Formula: see text].


1993 ◽  
Vol 18 (3) ◽  
pp. 206-214
Author(s):  
Patricia Dorval

Patrice Pavis offers an excellent analysis of ostension in his dictionary of theatre. Ostension (Latin ostendere, to show), he writes, is an essential feature of performance. The expression ‘theatrical ostension’ calls to mind quite naturally the notions of set, stage properties, body motions, and so on. In addition to purely visual elements, ostension also refers to verbal signals. But ostension is rarely total; one often sees only fragments of the dramatic world. Ostension is a relative notion modulated by elements that are not shown or not clearly shown. Synecdoche (the part for the whole) is one possible mode of ostension: the director suggests a complex reality by a characteristic detail: a crown for the King … Performance often resorts to metaphor: one element conjures up another. Pavis concludes that a stylistics or rhetoric of ostension would be a most precious tool for performance analysis.


1999 ◽  
Vol 08 (06) ◽  
pp. 773-787 ◽  
Author(s):  
BLAKE MELLOR

In [2], Bar-Natan used unitrivalent diagrams to show that finite type invariants classify string links up to homotopy. In this paper, I will construct the correct spaces of chord diagrams and unitrivalent diagrams for links up to homotopy. I will use these spaces to show that, far from classifying links up to homotopy, the only rational finite type invariants of link homotopy are the linking numbers of the components.


2002 ◽  
Vol 34 (4) ◽  
pp. 433-461 ◽  
Author(s):  
PAUL SILLITOE

What is the cost of living in the Papua New Guinea highlands? An answer is sought using a time and energy accounting approach. The subsistence regime of Wola-speaking highlanders, the subjects of this investigation, comprises three components. The principal one is horticulture: people clearing gardens from forest and grassland, with tuberous crops predominating, notably sweet potato. The second component comprises animal rearing, notably of domestic pigs. The third, and least important, is hunting and gathering for food in the forest. The calculated returns on investments in these subsistence domains vary considerably. Gardens return in their crops between ten and fifteen times the energy expended in cultivation. Pigs may also give a good return, of four to five times the energy invested in rearing them, if slaughtered when adult, but people regularly keep animals for years and may incur negative energy returns on their labour investments. This relates to the high cultural premium put on pigs. Foraging for food is also energetically costly, the Wola expending four times more energy on these activities than they gain in return from the food they secure. This analysis of energy gains and losses challenges the relative notion of affluence as applied to foragers, by reviewing it in the comparative context of subsistence horticulture.


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