The knot states considered in this work.
Localization effects in a “paraknot,” or a ring polymer constrained to visit the four corners of a square of side . For large , the most-probable distribution corresponds to a symmetric distribution of segments among the four loops. However, at small , the symmetry is broken, and the most-probable distribution assigns a large majority of segments to one of the loops. The paraknot has become localized in a small part of the ring.
Localization of knots in SARs in both swollen and theta states. is the mean value of the shortest linear subchain of a ring that exhibits the same knot state as the complete ring. Extrapolation implies that , or that knots are localized in sufficiently large rings.
Snapshot of a localized trefoil knot, . During annealing, the knot spontaneously and frequently localizes in a short section of the ring. Extrapolations imply that localization dominates in sufficiently large rings.
Localization of knots in ideal, Gaussian rings. Effective exponents in the relationship assume the indicated values in the interval for the seven knot states shown. Extrapolation again implies that , or that localization also occurs in IRs.
Distribution of trefoil knot lengths in ideal, Gaussian rings. The most-probable knot length is about 10 and independent of .
Radius-of-gyration scaling for self-avoiding, swollen knots. Dashed curves are the least-squares lines. Solid curves are the least-squares fits to Eq. (6).
Radius-of-gyration scaling for self-avoiding, theta-state knots. Dashed curves are the least-squares lines. Solid curves are the least-squares fits to Eq. (6).
Scaling of radius of gyration of swollen knots with knot complexity, measured as :diameter ratio (or “rope-length”) of the maximally inflated conformation. Values of are indicated.
Characteristics of the knots studied in this work. crossing number, crossing number of the maximally inflated knot, of the maximally inflated knot, and number of steps required for the knot on the simple-cubic lattice.
Evidence for knot localization in several knots; or length of the shortest open sub-chain that is found in the same knot state as the complete chain.
Effective metric exponents for swollen and theta knots. Slopes of the least-squares lines appearing in Figs. 7 and 8 for .
Least-squares fits of Eq. (6) for swollen knots.
Least-squares fits of Eq. (6) for theta-state knots.
Effective exponents for the dependence of on and .
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