^{1,a),b)}and Jack F. Douglas

^{2,a),c)}

### Abstract

We report calculations on three classes of knotted ringpolymers: (1) simple-cubic lattice self-avoiding rings (SARs), (2) “true” theta-state rings, i.e., SARs generated on the simple-cubic lattice with an attractive nearest-neighbor contact potential (-SARs), and (3) ideal, Gaussian rings. Extrapolations to large polymerization index imply knot localization in all three classes of chains. Extrapolations of our data are also consistent with conjectures found in the literature which state that (1) asymptotically for ensembles of random knots restricted to any particular knot state, including the unknot; (2) is universal across knot types for any given class of flexible chains; and (3) is equal to the standard self-avoiding walk (SAW) exponent for all three classes of chains (SARs, -SARs, and ideal rings). However, current computertechnology is inadequate to directly sample the asymptotic domain, so that we remain in a crossover scaling regime for all accessible values of . We also observe that , where is the “rope length” of the maximally inflated knot. This scaling relation holds in the crossover regime, but we argue that it is unlikely to extend into the asymptotic scaling regime where knots become localized.

We acknowledge the assistance of Professor Piotr Pieranski and Professor Sylvester Przybyl, Poznan University of Technology, Poznan, Poland, and of Professor Eric J. Rawdon, University of St. Thomas, St. Paul, Minnesota, USA in providing maximally-inflated conformations of knots and (−3,5,7)-pretzel.

I. INTRODUCTION

II. SIMULATION TECHNIQUES

III. KNOT TYPES EXAMINED IN THIS STUDY

IV. KNOT LOCALIZATION

V. DATA

VI. CONCLUSIONS

## Figures

The knot states considered in this work.

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 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.

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.

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.

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 .

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, 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).

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.

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.

## Tables

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.

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.

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 .

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 swollen knots.

Least-squares fits of Eq. (6) for theta-state knots.

Least-squares fits of Eq. (6) for theta-state knots.

Effective exponents for the dependence of on and .

Effective exponents for the dependence of on and .

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