KNOTTED 2-SPHERES IN TUBES IN Z<sup>4</sup>

Soteros, C E; Sumners, D. W.; Whittington, S G

doi:10.1142/s0218216512501167

Cited by 4 publications

(17 citation statements)

References 22 publications

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“…Note that the concatenation of two orientable (non-orientable) 2-manifolds yields an orientable (non-orientable) 2-manifold and the concatenation of an orientable and a non-orientable 2-manifold yields a non-orientable so, roughly speaking, the exponential growth rate is a non-decreasing function of the genus. If we take g = 0 and l = 2, the limit on the left-hand side exists [37,38,42] so that the exponential growth rate (if it exists) of projective planes in Z d , ⩾ d 4, is at least as great as that of spheres.…”

Section: Definitions and Some Preliminary Resultsmentioning

confidence: 99%

“…Slipknots have been observed in proteins [24] and have been studied using similar approaches [28]. Similar questions have been addressed about the higher dimensional analogue of knotting of 2-spheres in the four-dimensional lattice Z 4 [38], almost unknotted embeddings of graphs and surfaces [27], and linking of simple closed curves and surfaces [37]. Pattern theorems [15,23,26] have played an important role in many of these papers.…”

Section: Introductionmentioning

confidence: 85%

“…For the case of 2-spheres in a tube we have a pattern theorem [38] that implies that all except exponentially few such 2-spheres will have vertices in…”

Section: -Manifolds In a Tube In Z Dmentioning

confidence: 99%

“…The influence of bending rigidity on the behaviour of vesicles has also been investigated [19,33]. There are some results about the existence of an appropriate thermodynamic limit for embeddings of certain surfaces in lattices [8,16,18,20,37,38,42] and, for surfaces with boundary, it is known that the exponential growth rate is independent of the genus and of the fixed number of boundary curves [20]. In three dimensions the boundaries can be knotted and it is known that the exponential growth rate is independent of the knot type [21].…”

Section: Introductionmentioning

confidence: 99%

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Counting closed 2-manifolds in tubes in hypercubic lattices

Atapour¹,

Soteros²,

Sumners³

et al. 2015

J. Phys. A: Math. Theor.

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We consider closed 2-manifolds (2-manifolds without boundary) embedded in tubes in the hypercubic lattice, Z d , ⩾ d 3. For orientable 2-manifolds with fixed genus, ≠ d 4, we prove that the exponential growth rate is independent of the genus and we use this to prove a pattern theorem for manifolds with fixed genus. We prove a similar theorem for the non-orientable case for > d 4. If the genus is not fixed then we prove a pattern theorem and use this to show that 2-manifolds with genus less than any fixed number are exponentially rare so the typical genus increases with the size of the manifold. In four and higher dimensions we prove that orientable manifolds are exponentially rare and are dominated by non-orientable manifolds. In four dimensions, all except exponentially few 2-manifolds, both orientable and non-orientable, contain a local knotted (4, 2)-ball pair.

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Section: Definitions and Some Preliminary Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 85%

“…For the case of 2-spheres in a tube we have a pattern theorem [38] that implies that all except exponentially few such 2-spheres will have vertices in…”

Section: -Manifolds In a Tube In Z Dmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Counting closed 2-manifolds in tubes in hypercubic lattices

Atapour¹,

Soteros²,

Sumners³

et al. 2015

J. Phys. A: Math. Theor.

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show abstract

“…In another direction, random knot models may extend to knotted 2-spheres or other surfaces in a 4-sphere, and further to randomly embedded manifolds in higher dimensions (Soteros et al 2012;Atapour et al 2015).…”

Section: Random 3-manifoldsmentioning

confidence: 99%

Models of random knots

Even-Zohar

2017

J Appl. and Comput. Topology

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The study of knots and links from a probabilistic viewpoint provides insight into the behavior of ''typical'' knots, and opens avenues for new constructions of knots and other topological objects with interesting properties. The knotting of random curves arises also in applications to the natural sciences, such as in the context of the structure of polymers. We present here several known and new randomized models of knots and links. We review the main known results on the knot distribution in each model. We discuss the nature of these models and the properties of the knots they produce. Of particular interest to us are finite type invariants of random knots, and the recently studied Petaluma model. We report on rigorous results and numerical experiments concerning the asymptotic distribution of such knot invariants. Our approach raises questions of universality and classification of the various random knot models.

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