Topology and Geometry in Polymer Science pp 49-65 | Cite as

# Energy and Thickness of Knots^{*}

## Abstract

Knots in real physical systems, be they rope or DNA loops, have real physical properties; that is a truism. The behavior of *physical knots* depends on the topological types of the knots; that is an experimental observation. Mathematical differences between knot types and, more generally, the whole body of knot theory should help explain the physical behavior; that is a hope. If we make the three simplest knots out of similar “rope” having the same lengths Figure 1, we see that one kind of knot seems more “tight” or “compact” than another. Somehow this difference will manifest itself in physical systems and should be predictable from the topology.

### Keywords

Vortex Manifold Recombination Agarose Electrophoresis## Preview

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### References

- [Ad]C. Adams. B. Brennan, D. Greilsheimer and A. Woo,
*Stick numbers and**compositions of knots**and links*, J. Knot Theory and its Ramif., to appear.Google Scholar - [B]G. Buck, On
*the energy*and*length of a knot*, Talk in Special Session on Physical Knot Theory, Amer. Math. Soc.. Iowa City, March 1996.Google Scholar - [B3]G. Buck,
*Random knots and energy: elementary consideration*, J. Knot Theory and its Ramif. 3 (1994) (reprinted in book Random Knotting and Linking, K.C. Millett and D.W. Sumners (eds), World Scientific, 1994).Google Scholar - [B01]G. Buck and J. Orloff, A
*simple energy function for knots*, Topology and its Appl. 61 (1995), 205–214.MathSciNetCrossRefMATHGoogle Scholar - [B02]G. Buck and J. Orloff,
*Computing canonical conformations of knots*, Topology and its Appl. 51 (1993), 246–253.MathSciNetGoogle Scholar - [Br]K. Brakke (further devel, by J. Sullivan),
*SURFACE EVOLVER*, Geometry Center University of Minnesota, http://www.geom.urnri.edu/software - [BS]G. Buck and J. Simon, Knots
*as dynamical systems*, Topology and its Appl. 51 (1993), 229–246.MathSciNetCrossRefMATHGoogle Scholar - [BS2]G. Buck and J. Simon,
*Energy and length of knots*, Lectures at Knots ‘86, Proc. of summer 1996 Intl. Conf., Tokyo, S. Suzuki (ed.), World Scientific Publ. (1997), 219–234.Google Scholar - [BS3]G. Buck and J. Simon,
*Thickness and crossing*number*of*knots, announced in preceding conf., to appear, Topology and its Applications.Google Scholar - [Cr]N.J. Crisona, R. Kanaar, T.N. Gonzales, E.L. Zechiedrich, E.L. Klippei., and N.R. Cozzarelli,
*Processive**recombination by wild-type Gin and an enhancer-independent mutant. Insight into*the*mechanisms*of*recombination*and*strand exchange*,*J.*Mol. Biol. 243 (1994), 437–457.CrossRefGoogle Scholar - [Dia]Y. Diao
*Minimal knotted polygons on the cubic lattice*, J. Knot Theory and its Ramif. 2 (1993), 413–425.MathSciNetCrossRefMATHGoogle Scholar - [DC]P. Dröge and N.R. Cozzarelli
*Recombination*of*knotted substrates*by Tn3*resolvent*, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), 6062 6066.CrossRefGoogle Scholar - [DC2]P. Dröge and N.R. Cozzarelli,
*Topological**structure of DNA knots and cat manes*, Methods in Enzymology 212 (1992), 120–130.CrossRefGoogle Scholar - [De]F.B. Dean, A. Stastak, T. Koller, and N.R. Cozzarelli,
*Duplex*DNA*knots produced by escherichia colo topoisomerase I*, J. Biol. Chem. 260 (1985), 4975–4983.Google Scholar - [Deg]T. Deguchi and K. Tsurusaki, A
*statistical study*of random*knotting*using*the Vassiliev invariants*, J. Knot Theory and its Ramif. 3 (1994), 321–353 (reprinted in book Random Knotting and Linking, K.C. Millett and D.W. Sumners (eds), World Scientific, 1994).Google Scholar - [DEJ]Y. Diao, K. Ernst, and E.J.J. Vanrensburg,
*Energies*of*knots*(preprint, 3/95);*Knot energies by ropes*(preprint 4/96).Google Scholar - [Di]D. Dichmann, Y. Li, and J.H. Maddocks, Hamiltonian
*formulations and symmetries in rod mechanics*,Mathematical Approach to Biomolecular Structure and Dynamics (J.P. Mesirov, K. Schulten, and D.W. Sumners, ed.), Proc. of 1994 1MA Summer Program on Molecular Biology, IMA Volume no. 82, Springer-Verlag, 1996,71–113.Google Scholar - [DPS]Y. Diao, N. Pippenger, and D.W. Sumners, On random
*knots*,*J.*Knot Theory and its Ramif, 3 (1994) (reprinted in book Random Knotting and Linking, K.C. Millett and D.W. Sumners (eds), World Scientific, 1994).Google Scholar - [DSKC]F.B. Dean, A. Stasiak, T. Koller, and N.R. Cozzarelli, Duplex
*DNA*knots*produced by escherichia coli*topoisomerase*I*, J. Biological Chemistry 260 (1985), 4975–4983.Google Scholar - [FHW]M. Freedman, Z.-X. He, and Z. Wang
*Mains energy of knots and unknots*, Annals of Math. 139 (1994), 1–50.MathSciNetCrossRefMATHGoogle Scholar - [FW]H.L. Frisch and E. Vvasserman
*Chemical topology*, J. Am. Chem. Soc. 83 (1961), 3789–3795.CrossRefGoogle Scholar - [Fu]S. Fukuhara,
*Energy of*a*knot*, A Fete of Topology: Papers Dedicated to Itiro Tamura (Y.T. Matsumoto and S. Morita, ed.), Academic Press, New York, 1988,443–451.Google Scholar - [G]M. Gromov,
*Filling Riemannian manifolds*, J. Diff. Geom. 18 (1983), 1–147 (see p. 113).MathSciNetMATHGoogle Scholar - [Fle]J.E. Hearst and
*Y.*Shi,*The elastic rod provides*a*model for DNA and its functions*, Mathematical Approach to Biomolecular Structure and Dynamics (J.P. Mesirov, K. Schuhen, and D.W. Sumners, ed.), Proc. of 1994 IMA Summer Program on Molecular Biology, IMA Volume no. 82, Springer-Verlag, 1996,59–70.Google Scholar - [Hg]M. Huang, Univ. Illinois-Chicago,http://www.eecs.uic.eduk- mhuang/research.html, Program for visualizing and energy minimizing knots.
- [Hu]K. Hunt,
*KED*, University of Iowa,http://www.csaiiowa.edubhunti, Program for visualizing, manipulating, and energy minimizing polygonal knots. - [JvW]E.J. Janse Van Rensburg,
*Lattice invariants for knots*, Mathematical Approach to Biomolecular Structure and Dynamics (J.P. Mesirov, K. Schulten, and D.W. Sumners, ed.), Proc. of 1994 IMA Summer Program on Molecular Biology, IMA Volume no. 82, Springer-Verlag, 1996,11–20.Google Scholar - [JOS]E.J. Janse Van Rensburg, E. Orlandini, D.W. Sumners, M.C. Test and S. Whittington,
*Topology and geometry of biopolymers*, Mathematical Approach to Biomolecular Structure and Dynamics (J.P. Mesirov, K. Schulten, and D.W. Surnners, ed.), Proc. of 1994 IMA Summer Program on Molecular Biology, IMA Volume no. 82, Springer-Verlag, 1996,21–37.Google Scholar - [JW]E.J. Janse Van Rensburg and S. Whittington,
*The dimensions of knotted polygons*,*J.*Phys. A: Math. Gen. 24 (1991), 3935–3948.CrossRefGoogle Scholar - [JW2]E.J. Janse Van Rensburg and S. Whittington,
*The knot probability in lattice polygons*, J. Phys. A: Math. Gen 23 (1990), 3573–3590.CrossRefMATHGoogle Scholar - [JW3]E.J. Janse Van Rensburg and S. Whittington,
*The BFACF algorithm and knotted polygons*, J. Phys. A: Math. Gen. 24 (1991), 5553–5567.CrossRefMATHGoogle Scholar - [Jin]G.T. Jin and H.S. Kim,
*Polygonal knots*, Jour. Korean Math. Soc. 30 (1993), 371–383.MathSciNetMATHGoogle Scholar - [Jin2]G.T. Jin,
*Polygon indices and superbridge**indices of torus knots and links*, preprint 3/96.Google Scholar - [Ketc]R. Kanaar, A. Kuppel, E. Shekhtman, J.M. Dungan, R. Kahmann and N.R. Cozzarelli,
*Processive recombination by the phage Mu Gin system: Implications for the mechanisms**of DNA strand exchange*,*DNA site alignment*,*and enhancer action*, Cell 62 (1990), 353–366.CrossRefGoogle Scholar - [KB]V. Katritch, J. Bednar, D. Mtchoud, R. Scharein, J. Dubochet and A. Stasiak,
*Geometry and physics of knots*, Nature 384, Nov. 1996, 142–144.MathSciNetCrossRefGoogle Scholar - [KS]R. Kusner and J. Sullivan,
*McIbius energies for knots and links*,*surfaces and manifolds*, Geometry Center Research Report GCG64 (1993 rev. 1994), University of Minnesota, to appear in Proc. of 1993 Georgia Top. Cord..Google Scholar - [Ku]N. Kuiper, pers. corresp. with J. O’Hara, see [02]
*/oc cit.*Google Scholar - [L]R. Litherland,
*Thickness of knots*, Talk in Workshop on 3-Manifolds, Univ. Tennessee, 1992.Google Scholar - [L2]R. Litherland
^{•}, J. Simon, and O. Durumeric,*Thickness of knots*,Talk in AMS Special Session on Physical Knot Theory, Iowa City, 3/96.Google Scholar - [LCJ]H.A. Lim, M.T. Carroll, and E.J. Janse Van Rensburg,
*Electrophoresis of*knotted*DNA in a regular and a random electrophoretic medium*, Biomedical Modeling and Simulation (J. Eisenfeld, D.S. Levine and M. Witten, ed.), Elsevier Science Pub., 1992,213–223.Google Scholar - [Le]S.D. Levene and H. Tsen,
*Analysis*of*DNA knots and*catenanes by*agarosegel electrophoresis*, Protocols in DNA Topology and Topoisomerases, vol. I (M. Bjornsti and N. Osheroff, ed.), Humana Press, 1996.Google Scholar - [LJ]H.A. Lim and E.J. Janse Van Rensburg, A
*numerical simulation of electrophoresis of knotted*DNA, Supercomputer Computations Research Inst. Report FSU-SCRI-91–163 (1991).Google Scholar - [Lo]S. Lomonaco, The
*modern legacies of Thompson’s atomic*vortex theory*in classical electrodynamics*, The Interface of Knots and Physics (L. Kauffman, ed.), (AMS Short Course, Jan. 1995), American Mathematical Society, 1996,145 166.Google Scholar - [LSDR]R. Litherland, J. Simon, O. Durumeric, and E. Rawdon,
*Thickness of knots*(based on [L,Sil]), to appear in Topology and its Applic.Google Scholar - [Me]M. Meissen,
*Polygon knot*table, University of Iowa,http://www.math.uiowa.edu/~meissen/PLKnotTable.html. - [MS]K.C. Millett and D.W. Sumners(eds),
*Random Knotting and Linking*, World Scientific, 1994.Google Scholar - [Mo]H.K. Moffatt, The
*energy spectrum of knots*and*links*, Nature 347, Sept. 1999,367–369.CrossRefGoogle Scholar - [O1]J. O’hara,
*Energy of a knot*, Topology 30 (1991), 241–247.MathSciNetCrossRefMATHGoogle Scholar - [O2]J. O’hara,
*Family of energy*functionals*of knots*, Topology Appl. 48 (1992), 147–161.MathSciNetCrossRefMATHGoogle Scholar - [O3]J. O’hara,
*Energy functionals of knots*, Topology-Hawaii (K. H. Doverman), (Proc. of 1991 Conference), World Scientific, 1992,201–214 (Computer program by K. Ahara).Google Scholar - [O4]J. O’hara,
*Energy functionals of knots II*, Topology and its Appl. 56 (1994), 45–61.MathSciNetCrossRefMATHGoogle Scholar - [Ol]W. Olson, T. Westcott, J. Martino, G-H Liu, Computational
*studies of spatially constrained*DNA, Mathematical Approach to Biomolecular Structure and Dynamics (J.P. Mesirov, K. Schuhen, and D W Sumners, ed.), Proc. of 1994 IMA Summer Program on Molecular Biology, IMA Volume no. 82, Springer-Verlag, 1996,195–217.Google Scholar - [Pi]N. Pippenger,
*Knots in*random*walks*, Disc. Appl. Math. 25 (1989), 273–278.MathSciNetCrossRefMATHGoogle Scholar - [R]R. Randell, An
*elementary invariant*of*knots*, J. Knot Theory and its Ramif. 3 (1994), 279–286 (reprinted in book Random Knotting arid Linking, K.C. Millett and D.W. Sumners (eds), World Scientific, 1994).Google Scholar - [Rwl]E. Rawdon, The
*thickness page*, University of Iowa,http://www.math.uiowa.edu/rawdon/thick.html Preliminary data on program for finding thickest knots. - [Rw2]E. Rawdon,
*Thickness of Polygonal Knots*, Ph.D. Thesis, University of Iowa, August 1997.Google Scholar - [Ry]B.Y. Rybenkov, N.R. Cozzarelli, and A.V. Vologodskii, Probability
*of*DNA*knotting*and the effective diameter*of the*DNA*double helix*, Proc. Nat. Acad. Sci. 90 (1993), 5307–5311.CrossRefGoogle Scholar - [S]T. Schlick,
*Pursuing Laplace’s vision on modern computers*, Mathematical Approach to Biomolecular Structure and Dynamics (J.P. Mesirov, K. Schulten, and D.W. Summers, ed.), Proc. of 1994 IMA Summer Program on Molecular Biology, IMA Volume no. 82, Springer-Verlag, 1996, 219–247’.Google Scholar - [Sc]R. Scharein,
*Knot-Plot*,Univ. British Columbia, http://www.cs.ubc.ca/spider/sdiareini Program for drawing, visualizing, manipulating, and energy minimizing knots. - [Si1]J. Simon,
*Thickness of Knots*, Talk in Special Session on Knotting Phenomena in the Natural Sciences, American Mathematical Society, Santa Barbara, Nov. 1991;*Lecture**Notes on Physical Knot Theory*(notes by H. Naka), 1993, from summer 1991 course, Kwansei Gakuin University.Google Scholar - [Si2]J. Simon,
*Energy functions for polygonal knots*, J. Knot Theory and its Ramif. 3 (1994), 299–320 (reprinted in book Random Knotting and Linking, K.C. Millet t and D.W. Sumners (eds), World Scientific, 1994).Google Scholar - [Si3]J. Simon,
*Energy functions for knots: beginning to predict physical behavior*, Mathematical Approach to Biomolecular Structure and Dynamics (J.P. Mesirov, K. Schulten, and D W Sumners, ed.), Proc. of 1994 IMA Summer Program on Molecular Biology, IMA Volume no. 82, Springer- Verlag, 1996,39–58.Google Scholar - [Si4]J. Simon, papers available by anonymous ftp,http://www.nriath.uiowa.eduhjsimon/README.html.ISSCI S. Spengler
- [SSC]A. Stasiak, and N.R. Cozzarelli, The
*stereostructure**of knots and catenanes produced by phage*A*integrative recombination: implications for mechanism and DNA structure*, Cell 42 (1985), 325–334.CrossRefGoogle Scholar - [SK]A. Stasiak, V. Katritch, J. Bednar, D. Michoud, and J. Dubochet,
*Electrophoretic mobility of DNA knots*, Nature 384, Nov. 1996,122.MathSciNetCrossRefGoogle Scholar - [St]A. Stasiak,
*Ideal forms of**knots*, Talk in Special Session on Physical Knot Theory, Amer. Math. Soc., Iowa City, March 1996 (Revised paper to appear Dec. 96 Nature).Google Scholar - [SuW]D.W. Sumners and S. Whittington,
*Knots in self-avoiding walks*, J. Phys. A: Math. Gen. 21 (1988), 1689–1694.MathSciNetCrossRefMATHGoogle Scholar - [SW]S.Y. Shaw and J.C. Wang,
*Knotted*DNA*rings: probability of formation and resolution of the*two*chiral trefoils*, Science 260 (1993), 533–536.CrossRefGoogle Scholar - [SW2]S.Y. Shaw and J.C. Wang,
*DNA knot formation in*aqueous*solutions*, J. Knot Theory and its Ramif. 3 (1994), 287–298.MathSciNetCrossRefMATHGoogle Scholar - [Ta]M. Tabor and I. Klapper,
*Dynamics of twist and writhe and the modelingof bacterial fibers*, Mathematical Approach to Biomolecular Structure and Dynamics (J.R Mesirov, K. Schulten, and D.W. Sumners, ed.), Proc. of 1994 IMA Sununer Program on Molecular Biology, IMA Volume no, 82, Springer-Verlag, 1996,139–159.Google Scholar - [VLF]A.V. Vologodskii, A.V. Lukashin, M.D. Frank-Kamenetskii and V.V. Anshelevich,
*The knot probability in statistical mechanics of polymer chains*, Soy. Phys.-JETP 39 (1974), 1059–1063.MathSciNetGoogle Scholar - [Was]E. Wasserman,
*Chemical Topology*, Scientific American 207(5) (1962), 94–102.CrossRefGoogle Scholar - [WC]S. Wasserman and N.R. Cozzarelli, Supercoiled
*DNA-directed knotting by T4 topoisomerase*, J. Biol. Chem. 266 (1991), 73–95.Google Scholar - [Wu]Y.-Q. Wu,
*Ming*,University of Iowa, http://www.math.uiowa.edu/~wu/, Program for visualizing, manipulating, and energy minimizing polygonal knots.