The Topology of Knots

Methods have been devised for synthesis of even very complex DNA knots.185 186 Let s look briefly at the topology of knots. The three simple knots shown here have a chirality beyond that of the nucleotide... 

The above examples provide just a partial - and biased -illustration of chain dynamics. We hope, however, that it can be useful to show the very broad spectrum of interests which converge in polymers science from the topology of knots to how glues work. . . ... 

Metzler R, Hanke A (2005) Knots, bubbles, unwinding, and breathing probing the topology of DNA and other biomolecules. In Rieth M, Schommers W (eds) Handbook of theoretical and computational nanotechnology. American Scientific, California (in press)... 

Topological quantum field theory has become a fascinating and fashionable subject in mathematical physics. At present, the main applications of topological field theory are in mathematics (topology of low-dimensional manifolds) rather than in physics. Its application to the issue of classification of knots and links is one of the most interesting. To approach this problem, one usually tries to somehow encode the topology of a knot or link. As was first noted by Witten... 

In conclusion, knotted ligands and their complexes have new specific properties which originate from both the topology of the ligand backbone and the compactness and rigidity of the complexes made. 

ZEEMAN, ERIK CHRISTOPHER (1925-). Zeeman was an English mathematician. His doctoral work was in pure mathematics and he received his Ph D. in 1954 for a thesis on knots and all the algebra you need to actually prove the existence of knots. He did research in topology, which is a type of geometry that examines the properties of shapes in many dimensions. His best known work was in catastrophe theory. His work has consequences for a broad range of fields from weather to psychiatry. Zeeman also made contributions in the development of the chaos theory. 

In summary, failure to detect a rigidly achiral presentation does not mean that such a presentation cannot be found among the infinitely many presentations of a knot failure to interconvert enantiomorphous presentations by ambient isotopy does not exclude the possibility that an interconversion pathway can be found among the infinitely many pathways that are available and a palindromic knot polynomial does not necessarily mean that the knot is amphicheiral. Consequently, it may be impossible in certain cases to determine with complete certainty whether a knot is topologically chiral or not. The fundamental task of the theory of knots was stated over a hundred years ago by its foremost pioneer Given the number of its double points, to find all the essentially different forms which a closed curve can assume. 15 Yet to find invariants that will definitively determine whether or not a knot is chiral remains an unsolved problem to this day.63a Vassiliev invariants have been conjectured to be such perfect invariants.63b... 

In particular, the most obvious topological question, concerning the probability of knotting during random closure of polymer chain, cannot be answered using the Gauss invariant. 

Liquid extraction was used to make diastereomers, exploiting the high solubility of potassium triflate in water compared with the binaphthylphosphate salts. The two diastereomers have different solubilities and the (+) isomers of knot and anion crystallise together [49, 50], while the laevorotatory knot remains soluble. Counterion exchange with hexafluorophosphate gave the pure topological enantiomers. The optical rotatory power of the copper knots is very high At the sodium D-line (589 nm), the optical rotatory power was 7.000 mol 1 L dm They are beautiful molecules with a remarkable property ... 

A theorem proved by O. Gonzalez and J. Maddocks (Global curvature, thickness, and the ideal shapes of knots. Proceedings ofthe National Academy of Sciences, USA, 96 [1999], 4769) in the context of knot topologies shows the remarkable connection between the three-point recipe and the tube thickness. 

Sauvage et al. have elaborated special molecules with the topologies of catenated rings and knots [29]. The syntheses of these depend on metal coordination sites which direct the assembly of the components, but in general these catenanes and knots have few bonded connections. They are special because non-bonded sections are topologically trapped to coexist, thereby creating new opportunities for investigation of supramolecular relationships. A... 

Statistical mechanics of knots. To solve the problem of statistical mechanics of knots, one needs, first of all, a knot invariant. Indeed a closed chain can be unknotted or can form knots of different types. The very beginning of the table of knots is shown in Fig.8. However, an analytical expression for the knot in variant is unknown. Therefore, we had to use a computer and analgebraic invariant elaborated in the topological theory of knots. We found that the most convenient in variant was the Alexander polynomial (reviewed by Frank-Kamenetsku and Vologodskh, 1981 [24] and Vologodskh and Frank-Kamenetskii, 1992 [81]). 

And how about experiments on polymer knots The most important and, luckily, also the easiest subject of such experiments is double helical DNA. One nice experiment can de done using DNA with sticky ends - a long double helix with each chain extending at one end by 15 or so unpaired nucleotides beyond the counterpart chain. If the sequences of these extending pieces are complementary to each other, they will stick upon first collision due to the random fluctuations of the double helical coil. Can we then determine the topology of the product ... 

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