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According to this study and others which we cite below, it seems likely that knots in long chains are localized. Therefore, unlike typical global properties, some knot properties may be sensitive to the distinction between ideal and theta chains.
-
According to this study and others which we cite below, it seems likely that knots in long chains are localized. Therefore, unlike typical global properties, some knot properties may be sensitive to the distinction between ideal and theta chains.
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The notion of indeterminate knot state is meant to represent a situation which could be encountered with actual ropes or strings. Imagine two tangled ropes. In one case, at least one end of the rope is embedded in the tangle, while in the other, both ends are free and clear of the tangled portion of the rope. All reasonable individuals would agree that the latter rope has a definite knot state, but they might not agree about the definition of the knot state in the former case. To avoid any subjectivity or arbitrariness, we choose to leave the knot state undefined in the former situation.
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The notion of indeterminate knot state is meant to represent a situation which could be encountered with actual ropes or strings. Imagine two tangled ropes. In one case, at least one end of the rope is embedded in the tangle, while in the other, both ends are free and clear of the tangled portion of the rope. All reasonable individuals would agree that the latter rope has a definite knot state, but they might not agree about the definition of the knot state in the former case. To avoid any subjectivity or arbitrariness, we choose to leave the knot state undefined in the former situation.
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note
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It would not be inappropriate to refer to the localized knots discussed here as "tight knots." However, we point out that DeGennes has used the term "tight knot," [P. -G. DeGennes, Macromolecules MAMOBX 0024-9297 17, 703 (1984)] in what we believe is a different context. He considered knots in polymers that were "atomically tight" (our neologism, not his), or knots tightened by an externally applied tension sufficient to generate a unique conformation that is essentially the polymeric equivalent of the maximally inflated knot. Knot localization, in the present context, should be thought of as arising from an internal tension generated by the excluded volume interaction, but we do not envision that this tension is strong enough to force the knot into a unique conformation. He speculated that a sufficiently strong tension might create stable tight knots. Molecular dynamics calculations indicate, however, that room-temperature thermal motion is adequate to loosen even the tightest knots, and that a knotted polymer strand breaks under tension before forming a stable tight knot 10.1021/ma00134a030 [M. L. Mansfield, Macromolecules MAMOBX 0024-9297 31, 4030 (1998)]. 10.1021/ma980013l
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