Relative Frequencies of Alternating and Nonalternating Prime Knots and Composite Knots in Random Knot Spaces

References

• [Adams 04] C. C. Adams. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. Providence, RI: American Mathematical Society, 2004.
   Google Scholar
• [Arsuaga et al. 02] J. Arsuaga, M. Vázquez, S. Trigueros, De Witt Sumners, and J. Roca. “Knotting Probability of DNA Molecules Confined in Restricted Volumes: DNA Knotting in Phage Capsids.” Proc. Natl. Acad. Sci. USA 99 (2002), 5373–5377.
   PubMed Web of Science ®Google Scholar
• [Ashton et al. 11] T. Ashton, J. Cantarella, M. Piatek, and E. J. Rawdon. “Knot Tightening by Constrained Gradient Descent.” Exp. Math. 20: 1 (2011), 57–90.
   Web of Science ®Google Scholar
• [Brown et al. 01] L. D. Brown, T. Tony Cai, and A. DasGupta. “Interval Estimation for a Binomial Proportion.” Stat. Sci. 16: 2 (2001), 101–117.
   Web of Science ®Google Scholar
• [Buck and Simon 99] G. Buck and J. Simon. “Thickness and Crossing Number of Knots.” Topol. Appl. 91: 3 (1999), 245–257.
   Web of Science ®Google Scholar
• [Burde et al. 14] G. Burde, H. Zieschang, and M. Heusener. Knots, volume 5 of De Gruyter Studies in Mathematics. Berlin: De Gruyter, extended edition, 2014.
   Google Scholar
• [Cantarella et al. 14] J. Cantarella, T. Deguchi, and C. Shonkwiler. “Probability Theory of Random Polygons from the Quaternionic Viewpoint.” Commun. Pure Appl. Math. 67: 10 (2014), 1658–1699.
   Web of Science ®Google Scholar
• [Cantarella et al. 02] J. Cantarella, R. B. Kusner, and J. M. Sullivan. “On the Minimum Ropelength of Knots and Links.” Invent. Math. 150: 2 (2002), 257–286.
   Web of Science ®Google Scholar
• [Cantarella and Shonkwiler 13] J. Cantarella and C. Shonkwiler. “The Symplectic Geometry of Closed Equilateral Random Walks in 3-Space.” Ann. Appl. Probab. 26: 1 (2016), 549–596.
   Web of Science ®Google Scholar
• [Diao 95] Y. Diao. “The Knotting of Equilateral Polygons in R3.” J. Knot Theory Ramifications 4: 2 (1995), 189–196.
   Web of Science ®Google Scholar
• [Diao and Ernst 98] Y. Diao and C. Ernst. “The Complexity of Lattice Knots.” Topol. Appl. 90: 1–3 (1998), 1–9.
   Web of Science ®Google Scholar
• [Diao et al. 97] Y. Diao, C. Ernst, and E. J. Janse van Rensburg. “Knot Energies by Ropes.” J. Knot Theory Ramifications 6: 6 (1997), 799–807.
   Web of Science ®Google Scholar
• [Diao et al. 14a] Y. Diao, C. Ernst, A. Montemayor, E. J. Rawdon, and U. Ziegler. “The Knot Spectrum of Confined Random Equilateral Polygons.” Mol. Math. Biol. 2: 1 (2014), 19–33.
   Google Scholar
• [Diao et al. 16] Y. Diao, C. Ernst, A. Montemayor, E. J. Rawdon, and U. Ziegler. “The Knot Spectrum of Random Knot Spaces.” Preprint, 2016.
   Google Scholar
• [Diao et al. 11] Y. Diao, C. Ernst, A. Montemayor, and U. Ziegler. “Generating Equilateral Random Polygons in Confinement.” J. Phys. A: Math. Theor. 44 (2011), 405202.
   Web of Science ®Google Scholar
• [Diao et al. 12] Y. Diao, C. Ernst, A. Montemayor, and U. Ziegler. “Generating Equilateral Random Polygons in Confinement II.” J. Phys. A: Math. Theor. 45 (2012), 275203.
   Web of Science ®Google Scholar
• [Diao et al. 12] Y. Diao, C. Ernst, A. Montemayor, and U. Ziegler. “Generating Equilateral Random Polygons in Confinement III.” J. Phys. A: Math. Theor. 45 (2012), 465003.
   Web of Science ®Google Scholar
• [Diao et al. 15] Y. Diao, C. Ernst, S. Saarinen, and U. Ziegler. “Generating Random Walks and Polygons with Stiffness in Confinement.” J. Phys. A: Math. Theor. 48: 9 (2015), 095202.
   Web of Science ®Google Scholar
• [Diao et al. 14b] Y. Diao, C. Ernst, and U. Ziegler. “Random Walks and Polygons in Tight Confinement.” J. Phys. Conf. Ser. 544: 1 (2014), 012017.
   Google Scholar
• [Diao et al. 94] Y. Diao, N. Pippenger, and De Witt Sumners. “On Random Knots.” J. Knot Theory Ramifications 3: 3 (1994), 419–429. Random knotting and linking (Vancouver, BC, 1993).
   Google Scholar
• [Ewing and Millett 97] B. Ewing and K. C. Millett. “Computational Algorithms and the Complexity of Link Polynomials.” In Progress in Knot Theory and Related Topics, volume 56 of Travaux en Cours, edited by M. Boileau, M. Domergue, Y. Mathieu, and K. Millett, pp. 51–68. Paris: Hermann, 1997.
   Google Scholar
• [Freedman et al. 94] M. H. Freedman, Z.-X. He, and Z. Wang. “Möbius Energy of Knots and Unknots.” Ann. Math. (2) 139: 1 (1994), 1–50.
   Web of Science ®Google Scholar
• [Gonzalez and Maddocks 99] O. Gonzalez and J. H. Maddocks. “Global Curvature, Thickness, and the Ideal Shapes of Knots.” Proc. Natl. Acad. Sci. USA 96: 9 (1999), 4769–4773.
   PubMed Web of Science ®Google Scholar
• [Hickford et al. 06] J. Hickford, R. Jones, S. Courrech Du Pont, and J. Eggers. “Knotting Probability of a Shaken Ball-Chain.” Phys. Rev. E 74: 5 (2006), 052101.
   Web of Science ®Google Scholar
• [Hoste and Thistlethwaite] J. Hoste and M. Thistlethwaite. Knotscape. Available online (http://www.math.utk.edu/~morwen/knotscape.html.) Program for computing topological information about knots.
   Google Scholar
• [Hoste et al. 98] J. Hoste, M. Thistlethwaite, and J. Weeks. “The First 1,701,936 Knots.” Math. Intell. 20: 4 (1998), 33–48.
   Web of Science ®Google Scholar
• [Katritch et al. 96] V. Katritch, J. Bednar, D. Michoud, R. G. Scharein, J. Dubochet, and A. Stasiak. “Geometry and Physics of Knots.” Nature 384 (1996), 142–145.
   Web of Science ®Google Scholar
• [Kawauchi 96] A. Kawauchi. A Survey of Knot Theory. Basel: Birkhäuser Verlag, 1996.
   Google Scholar
• [The Knot Atlas 17] The Knot Atlas. Available online (http://katlas.org), 2017.
   Google Scholar
• [Kusner and Sullivan 97] R. B. Kusner and J. M. Sullivan. “Möbius energies for Knots and Links, Surfaces and Submanifolds.” In Geometric Topology (Athens, GA, 1993), volume 2 of AMS/IP Stud. Adv. Math., pp. 570–604. Providence, RI: Amer. Math. Soc., 1997.
   Google Scholar
• [Raymond Lickorish 97] W. B. Raymond Lickorish. An Introduction to Knot Theory, volume 175 of Graduate Texts in Mathematics. New York: Springer-Verlag, 1997.
   Google Scholar
• [Litherland et al. 99] R. A. Litherland, J. Simon, O. Durumeric, and E. Rawdon. “Thickness of Knots.” Topol. Appl. 91: 3 (1999), 233–244.
   Web of Science ®Google Scholar
• [Livingston 93] C. Livingston. “Knot Theory.” In Carus Mathematical Monographs, vol. 24. Washington, DC: Mathematical Association of America, 1993.
   Google Scholar
• [Micheletti et al. 11] C. Micheletti, D. Marenduzzo, and E. Orlandini. “Polymers with Spatial or Topological Constraints: Theoretical and Computational Results.” Phys. Rep. 504: 1 (2011), 1–73.
   Google Scholar
• [Micheletti et al. 06] C. Micheletti, D. Marenduzzo, E. Orlandini, and De Witt Sumners. “Knotting of Random Ring Polymers in Confined Spaces.” J. Chem. Phys. 124: 6 (2006), 064903.
   PubMed Web of Science ®Google Scholar
• [Millett and Rawdon 03] K. C. Millett and E. J. Rawdon. “Energy, Ropelength, and Other Physical Aspects of Equilateral Knots.” J. Comput. Phys. 186: 2 (2003), 426–456.
   Web of Science ®Google Scholar
• [Pippenger 89] N. Pippenger. “Knots in Random Walks.” Discrete Appl. Math. 25: 3 (1989), 273–278.
   Web of Science ®Google Scholar
• [Raymer and Smith 07] D. M. Raymer and D. E. Smith. “Spontaneous Knotting of an Agitated String.” Proc. Natl. Acad. Sci. USA 104: 42 (2007), 16432–16437.
   PubMed Web of Science ®Google Scholar
• [Sumners and Whittington 88] D. W. Sumners and S. G. Whittington. “Knots in Self-Avoiding Walks.” J. Phys. A 21: 7 (1988), 1689–1694.
   Google Scholar
• [Sundberg and Thistlethwaite 98] C. Sundberg and M. Thistlethwaite. “The Rate of Growth of the Number of Prime Alternating Links and Tangles.” Pac. J. Math. 182: 2 (1998), 329–358.
   Web of Science ®Google Scholar
• [Thistlethwaite] M. Thistlethwaite. “Unraveller.” Code provided through private communication.
   Google Scholar