Knots definition

Definitions of the commonly measured tensile properties are as follows Unear density (tex) is the weight in grams of 1000 m of yam. Tenacity is the tensile stress at break and is expressed in force-per-unit linear density of unstrained specimen, N /tex. Knot tenacity is the tensile stress required to mpture a single strand of yam with an overhand knot tied in the segment of sample between the testing clamps. It is expressed as force-per-unit linear density and is an approximate measure of the britdeness of the yam. Toop tenacity is the tensile stress required to mpture yam when one strand of yam is looped through... 

Our definitions of the stereoisomeric center, line, and plane all stipulate the existence of bonds between the ligating element and its ligands. The exclusive use of these elements limits our analysis to classical stereochemistry and thus does not encompass the so-called topological isomerism (47) of interlocked rings—catenanes (48)—or of knots. As there is no bond between the rings of the catenanes we cannot expect to handle such compounds with a system based on connectedness. At the present stage of development, this limitation in scope... 

This definition is related to the difference between left- and right-handed photons because B(3) switches sign between left and right circularly polarized electromagnetic radiation. Therefore, H and B(3) constitute electromagnetic helicities of a knot, and there is also a link between B(3) and the Sachs theory [1], as shown in the review [6] by Evans, linking 0(3) electrodynamics and the Sachs 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... 

Thus, it should be stressed that the mathematical topological theory investigates, as a rule, the problems of classification of knots and links, the construction of topological invariants, definitions of topological classes, etc. whereas the fundamental physical problem in the theory of topological properties of polymer chains is the determination of the entropy, S = In Z with the fixed topological state of chains. Both these problems are very difficult, but important. 

We have seen how elegantly transition metals can template the formation of knots, but what about Nature s favourite templating interaction, the hydrogen bond A remarkably efficient molecular trefoil knot synthesis based on this interaction was reported by Vogtle and co-workers, who made a knotane in 20% yield [39]. This amazing route (Fig. 11) was uncovered serendipitously during the synthesis of catenanes. The crystal structure of the compound was the definitive proof for the structure, because neither NMR nor mass spectrometry could tell it apart conclusively from the macrocycles that are also formed. 

The recursive rules for the construction of the Jones polynomial VK(t) for an arbitrary knot K are shown in Figure 3.6. By definition, the Jones polynomial Vjj(t) of the unknot U is 1,... 

The spectrum emitted by an atom presumably is related to the structure of the atom. Until 1913, attempts to relate the spectrum to a definite atomic model were unsuccessful. By 1913 it was known that the atom had a positively charged nucleus, but the nuclear model of Rutherford was unstable according to classical electromagnetic theory. This Gordian knot was cut by Niels Bohr in 1913. 

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