Knotting statistics for polygons in lattice tubes. (English) Zbl 1509.82131
Summary: We study several related models of self-avoiding polygons in a tubular subgraph of the simple cubic lattice, with a particular interest in the asymptotics of the knotting statistics. Polygons in a tube can be characterised by a finite transfer matrix, and this allows for the derivation of pattern theorems, calculation of growth rates and exact enumeration. We also develop a static Monte Carlo method which allows us to sample polygons of a given size directly from a chosen Boltzmann distribution.
Using these methods we accurately estimate the growth rates of unknotted polygons in the \(2\times1\times\infty\) and \(3\times1\times\infty\) tubes, and confirm that these are the same for any fixed knot-type \(K\). We also confirm that the entropic exponent for unknots is the same as that of all polygons, and that the exponent for fixed knot-type \(K\) depends only on the number of prime factors in the knot decomposition of \(K\). For the simplest knot-types, this leads to a good approximation for the polygon size at which the probability of the given knot-type is maximized, and in some cases we are able to sample sufficiently long polygons to observe this numerically.
Using these methods we accurately estimate the growth rates of unknotted polygons in the \(2\times1\times\infty\) and \(3\times1\times\infty\) tubes, and confirm that these are the same for any fixed knot-type \(K\). We also confirm that the entropic exponent for unknots is the same as that of all polygons, and that the exponent for fixed knot-type \(K\) depends only on the number of prime factors in the knot decomposition of \(K\). For the simplest knot-types, this leads to a good approximation for the polygon size at which the probability of the given knot-type is maximized, and in some cases we are able to sample sufficiently long polygons to observe this numerically.
MSC:
| 82M31 | Monte Carlo methods applied to problems in statistical mechanics |
| 82B30 | Statistical thermodynamics |
| 80M31 | Monte Carlo methods applied to problems in thermodynamics and heat transfer |
| 57K14 | Knot polynomials |
Software:
KnotPlotReferences:
| [1] | Janse van Rensburg E J 2015 The Statistical Mechanics of Interacting Walks, Polygons, Animals, and Vesicles 2nd edn (Oxford: Oxford University Press) · Zbl 1314.82019 · doi:10.1093/acprof:oso/9780199666577.001.0001 |
| [2] | Vanderzande C 1998 Lattice Models of Polymers (Cambridge: Cambridge University Press) · Zbl 0922.76004 · doi:10.1017/CBO9780511563935 |
| [3] | Guttmann A J (ed) 2009 Polygons, Polyominoes and Polycubes(Lecture Notes in Physics vol 775) (Berlin: Springer) · Zbl 1162.82005 · doi:10.1007/978-1-4020-9927-4 |
| [4] | Whittington S G 1982 Statistical Mechanics of Polymer Solutions and Polymer Adsorption(Advances in Chemical Physics vol 51) (New York: Wiley) pp 1-48 |
| [5] | Guttmann A J and Whittington S G 1978 Two-dimensional lattice embeddings of connected graphs of cyclomatic index two J. Phys. A: Math. Gen.11 721 · doi:10.1088/0305-4470/11/4/013 |
| [6] | Whittington S G and Valleau J P 1969 Estimation of the number of self-avoiding lattice polygons J. Chem. Phys.50 4686-90 · doi:10.1063/1.1670956 |
| [7] | Sumners D W and Whittington S G 1988 Knots in self-avoiding walks J. Phys. A: Math. Gen.21 1689 · Zbl 0659.57003 · doi:10.1088/0305-4470/21/7/030 |
| [8] | Delbrück M and Fuller F B 1962 Knotting problems in biology Mathematical Problems in the Biological Sciences(Proc. of Symp. in Applied Mathematics vol 14) ed R E Bellman (Providence, RI: American Mathematical Society) pp 55-68 · doi:10.1090/psapm/014/9958 |
| [9] | Frisch H L and Wasserman E 1961 Chemical topology J. Am. Chem. Soc.83 3789-95 · doi:10.1021/ja01479a015 |
| [10] | Lim N C H and Jackson S E 2015 Molecular knots in biology and chemistry J. Phys.: Condens. Matter27 354101 · doi:10.1088/0953-8984/27/35/354101 |
| [11] | Orlandini E 2018 Statics and dynamics of DNA knotting J. Phys. A: Math. Theor.51 053001 · Zbl 1385.92036 · doi:10.1088/1751-8121/aa9a4c |
| [12] | Janse van Rensburg E J and Whittington S G 1990 The knot probability in lattice polygons J. Phys. A: Math. Gen.23 3573 · Zbl 0709.57001 · doi:10.1088/0305-4470/23/15/028 |
| [13] | Katritch V, Olson W K, Vologodskii A, Dubochet J and Stasiak A 2000 Tightness of random knotting Phys. Rev. E 61 5545-9 · doi:10.1103/PhysRevE.61.5545 |
| [14] | Shimamura M K and Deguchi T 2000 Characteristic length of random knotting for cylindrical self-avoiding polygons Phys. Lett. A 274 184-91 · Zbl 1065.60501 · doi:10.1016/S0375-9601(00)00545-4 |
| [15] | Uehara E and Deguchi T 2015 Characteristic length of the knotting probability revisited J. Phys.: Condens. Matter27 354104 · doi:10.1088/0953-8984/27/35/354104 |
| [16] | Uehara E and Deguchi T 2017 Knotting probability of self-avoiding polygons under a topological constraint J. Chem. Phys.147 094901 · doi:10.1063/1.4996645 |
| [17] | Baiesi M and Orlandini E 2012 Universal properties of knotted polymer rings Phys. Rev. E 86 031805 · doi:10.1103/PhysRevE.86.031805 |
| [18] | Cheston M A, McGregor K, Soteros C E and Szafron M L 2014 New evidence on the asymptotics of knotted lattice polygons via local strand-passage models J. Stat. Mech. P02014 · Zbl 1456.82946 · doi:10.1088/1742-5468/2014/02/p02014 |
| [19] | Janse van Rensburg E J 2008 Thoughts on lattice knot statistics J. Math. Chem.45 7 · Zbl 1258.82003 · doi:10.1007/s10910-008-9364-9 |
| [20] | Janse van Rensburg E J and Rechnitzer A 2011 On the universality of knot probability ratios J. Phys. A: Math. Theor.44 162002 · Zbl 1216.82018 · doi:10.1088/1751-8113/44/16/162002 |
| [21] | Orlandini E, Tesi M C, Janse van Rensburg E J and Whittington S G 1998 Asymptotics of knotted lattice polygons J. Phys. A: Math. Gen.31 5953 · Zbl 0953.82031 · doi:10.1088/0305-4470/31/28/010 |
| [22] | Yao A, Matsuda H, Tsukahara H, Shimamura M K and Deguchi T 2001 On the dominance of trivial knots among saps on a cubic lattice J. Phys. A: Math. Gen.34 7563 · Zbl 0982.82015 · doi:10.1088/0305-4470/34/37/310 |
| [23] | Whittington S G 2018 Knot probabilities for equilateral random polygons in R3 |
| [24] | Orlandini E and Whittington S G 2007 Statistical topology of closed curves: some applications in polymer physics Rev. Mod. Phys.79 611-42 · Zbl 1205.82153 · doi:10.1103/RevModPhys.79.611 |
| [25] | Micheletti C, Marenduzzo D and Orlandini E 2011 Polymers with spatial or topological constraints: theoretical and computational results Phys. Rep.504 1-73 · Zbl 1211.82070 · doi:10.1016/j.physrep.2011.03.003 |
| [26] | Whittington S G 1983 Self-avoiding walks with geometrical constraints J. Stat. Phys.30 449-56 · doi:10.1007/BF01012318 |
| [27] | Hammersley J M and Whittington S G 1985 Self-avoiding walks in wedges J. Phys. A: Math. Gen.18 101 · doi:10.1088/0305-4470/18/1/022 |
| [28] | Soteros C E and Whittington S G 1988 Polygons and stars in a slit geometry J. Phys. A: Math. Gen.21 L857 · doi:10.1088/0305-4470/21/9/031 |
| [29] | Whittington S G and Soteros C E 1991 Polymers in slabs, slits, and pores Isr. J. Chem.31 127-33 · doi:10.1002/ijch.199100014 |
| [30] | Soteros C E, Sumners D W and Whittington S G 1999 Linking of random p -spheres in Zd J. Knot Theory Ramifications08 49-70 · Zbl 0933.57021 · doi:10.1142/S0218216599000067 |
| [31] | Tesi M C, Janse van Rensburg E J, Orlandini E and Whittington S G 1994 Knot probability for lattice polygons in confined geometries J. Phys. A: Math. Gen.27 347 · Zbl 0820.60101 · doi:10.1088/0305-4470/27/2/019 |
| [32] | Tesi M C, Janse van Rensburg E J, Orlandini E and Whittington S G 1998 Topological entanglement complexity of polymer chains in confined geometries Topology and Geometry in Polymer Science ed S G Whittington et al (New York: Springer) pp 135-57 · Zbl 0940.57012 · doi:10.1007/978-1-4612-1712-1_11 |
| [33] | Soteros C E 1998 Knots in graphs in subsets of Z3 Topology and Geometry in Polymer Science ed S G Whittington et al (New York: Springer) pp 101-33 · Zbl 0917.57004 · doi:10.1007/978-1-4612-1712-1_10 |
| [34] | Beaton N R, Eng J W, Ishihara K, Shimokawa K and Soteros C E 2018 Characterising knotting properties of polymers in nanochannels Soft Matter14 5775-85 · doi:10.1039/C8SM00734A |
| [35] | Atapour M, Beaton N R, Eng J W, Ishihara K, Shimokawa K, Soteros C E and Vazquez M 2019 Unknotting operations on 4-plat diagrams and the entanglement statistics of polygons in a lattice tube (in preparation) |
| [36] | Beaton N R, Eng J W and Soteros C E 2016 Polygons in restricted geometries subjected to infinite forces J. Phys. A: Math. Theor.49 424002 · Zbl 1353.82029 · doi:10.1088/1751-8113/49/42/424002 |
| [37] | Atapour M, Soteros C E and Whittington S G 2009 Stretched polygons in a lattice tube J. Phys. A: Math. Theor.42 322002 · Zbl 1179.82020 · doi:10.1088/1751-8113/42/32/322002 |
| [38] | Soteros C E and Whittington S G 1989 Lattice models of branched polymers: effects of geometrical constraints J. Phys. A: Math. Gen.22 5259-70 · Zbl 0721.05060 · doi:10.1088/0305-4470/22/24/014 |
| [39] | Kloczkowski A and Jernigan R L 1998 Transfer matrix method for enumeration and generation of compact self-avoiding walks. II. Cubic lattice J. Chem. Phys.109 5147-59 · doi:10.1063/1.477129 |
| [40] | Eng J W 2014 Self-avoiding polygons in (L,M)-tubes Master’s Thesis University of Saskatchewan |
| [41] | Atapour M 2008 Topological entanglement complexity of systems of polygons and walks in tubes PhD Thesis University of Saskatchewan |
| [42] | Soteros C E, Sumners D W and Whittington S G 1992 Entanglement complexity of graphs in Z3 Math. Proc. Camb. Phil. Soc.111 75 · Zbl 0769.57007 · doi:10.1017/S0305004100075174 |
| [43] | Alm S E and Janson S 1990 Random self-avoiding walks on one-dimensional lattices Stoch. Models6 169-212 · Zbl 0702.60063 · doi:10.1080/15326349908807144 |
| [44] | Scharein R, Ishihara K, Arsuaga J, Diao Y, Shimokawa K and Vazquez M 2009 Bounds for the minimum step number of knots in the simple cubic lattice J. Phys. A: Math. Theor.42 475006 · Zbl 1185.82070 · doi:10.1088/1751-8113/42/47/475006 |
| [45] | Janse van Rensburg E J 2009 Monte Carlo methods for the self-avoiding walk J. Phys. A: Math. Theor.42 323001 · Zbl 1179.82074 · doi:10.1088/1751-8113/42/32/323001 |
| [46] | Scharein R 2018 The KnotPlot Site (www.knotplot.com) (Accessed: 30 November 2018) |
| [47] | Stolz R, Yoshida M, Brasher R, Flanner M, Ishihara K, Sherratt D J, Shimokawa K and Vazquez M 2017 Pathways of DNA unlinking: a story of stepwise simplification Sci. Rep.7 12420 · doi:10.1038/s41598-017-12172-2 |
| [48] | Williams T et al 2018 gnuplot (www.gnuplot.info) (Accessed: 30 November 2018) |
| [49] | Ishihara K, Pouokam M, Suzuki A, Scharein R, Vazquez M, Arsuaga J and Shimokawa K 2017 Bounds for minimum step number of knots confined to tubes in the simple cubic lattice J. Phys. A: Math. Theor.50 215601 · Zbl 1383.92017 · doi:10.1088/1751-8121/aa6a4f |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
