In this talk, describing joint work with Rob Kusner (UMASS) and John Sullivan (UIUC), we present a new family of bounds for the ropelength of knots and links. In some circumstances, these bounds are sharp, allowing us to construct a family of tight links. In others, we can obtain improved bounds for ropelength in terms of linking numbers, and "overcrossing numbers", and eventually improve the best known lower bounds for the ropelength of a nontrivial knot.

The global radius of curvature of a space curve is introduced. This function is related to, but distinct from, the standard local radius of curvature, and is connected to various physically appealing properties of a curve. In particular, the global radius of curvature function provides a concise characterization of the thickness of a curve, and of certain ideal shapes of knots as have been investigated within the context of DNA.

Spaces of polygonal knots, subject to specified constraints such as the number of nondegenerate edges or the requirement of having fixed edge lengths, provide the context within which it is natural to study configurations which are ideal with respect to a variety of natural physically motivated constraints. For knots with relatively few vertices the high dimensionality and complexity of the knot space makes analytical investigations impracticable. In this paper we will discuss the methods and the results of a Markov Chain Monte Carlo investigation of several fundamental problems concerning ideal knot configurations.

In certain Hamiltonian field theory models knots and links can be constructed as solitonic solutions to the classical equations of motion. This enables the study of various physical properties of knotlike configurations from first principles. Here we show how the pertinent field theory models are constructed, and review the present status of numerical studies of various knotted and linked configurations. We also explain how field theory models can be employed to study interaction processes including the splitting and joining of knotted configurations and describe the topological rules that govern the interaction vertex. Finally, we describe a number of physical scenarios where knotted solitons can be realized.

Remi Langevin proposed a new knot energy functional, i.e. a functional on the space of knots that blows up if a knot degenerates to have double points, from integral geometric viewpoint using conformal geometry. His functional is defined by the volume of spheres that intersects a given knot at more than 4 points, where the space of spheres is identified with the quadric hypersurface in R^5 which is endowed with Lorentz metric. The formulation and the relation to the energy E of knots, which is also conformal invariant, will be discussed.

Ideal geometric representations of knots (ideal knots) are defined as minimal length trajectories of uniform diameter tubes forming a given type of knot[1,2]. Ideal knots showed interesting relations between each other and turned out to be good predictors of certain average properties of randomly distorted knotted polymers[1-6]. More recently we demonstrated that writhe of ideal knots seems to be quantized whereby each crossing of the torus type introduces the writhe contribution of 10/7 (or -10/7) while each crossing of the twist type introduces the writhe contribution of 4/7 (or -4/7)[7]. This observation allowed us to propose a new topological invariant that predicts the writhe of ideal knots or the average writhe of random knots of a given type just from minimal diagrams of the corresponding knots or links[7,8].
References:
[1] V. Katritch, J. Bednar, D. Michoud, D., R. G., Scharein, J. Dubochet & A. Stasiak, Geometry and physics of knots. Nature, 384 (1996), pp. 142-145.
[2] A. Stasiak, J. Dubochet, V. Katritch & P. Pieranski, Ideal knots and their relation to the physics of real knots. in: Ideal Knots, A. Stasiak, V. Katritch & L. H. Kauffman (eds), World Scientific Publishing, Singapore, 1998, pp. 1-19
[3] A. Stasiak, V. Katritch, J. Bednar, D. Michoud & J. Dubochet, Electrophoretic mobility of DNA knots. Nature, 384 (1996), pp. 122.
[4] V. Katritch, W.K. Olson, P. Pieranski, J. Dubochet & A. Stasiak, Properties of ideal composite knots. Nature, 388 (1997), pp. 148-151.
[5] A. Vologodskii, N. Crisona, B. Laurie, P. Pieranski, V. Katritch, J. Dubochet & A. Stasiak, Sedimentation and electrophoretic migration of DNA knots and catenanes. J. Mol. Biol. 278 (1998), pp. 1-3.
[6] B. Laurie, V. Katritch, J. Sogo, T. Koller, J. Dubochet & A. Stasiak, Geometry and physics of catenanes applied to study DNA replication. Biophysical Journal, 74 (1998), pp. 2815-2822.
[7] A. Stasiak, Quantum-like properties of knots and links. in Knots in Hellas '98, C. Gordon, V.F.R. Jones, L. Kauffman, S. Lambropolou & J.H. Przytycki (eds), World Scientific, Singapore, 2000, to appear.
[8 ] C. Cerf & A. Stasiak, A topological invariant to predict the 3-D writhe of ideal configuration of knots and links. Proc. Natl Acad. Sci. USA. (2000), to appear.

For a model of random polygons with N nodes, we define the knotting probability of a knot K by the probability that a given random polygon is equivalent to the knot K . We introduce a model of self-avoiding polygons which describes knotted DNAs with an effective diameter d and of N Kuhn lengths. Then, we discuss the N -dependence of the knotting probability of the model for different diameters, through numerical simulations using knot invariants. In particular, we analyze the knotting probabilities of small-sized polygons from the viewpoint of the large N scaling behaviors. Our study might give some suggestions on how to interpret DNA experiments in order to detect the large N behaviors of random notting probability.

In research of Bernard D. Coleman, Irwin Tobias, and the speaker on the theory of the idealized elastic rod model for DNA exact analytical representations have been obtained for equilibrium configurations of DNA segments showing both isolated points and intervals of self-contact. Criteria have been derived for determining whether an equilibrium configuration is stable in the sense that it gives a strict local minimum to elastic energy. In recent joint research with Coleman these results have been applied to knotted DNA plasmids, with emphasis placed on the dependence on excess link of minimum elastic energy configurations of DNA torus knots. Among the results to be presented are: the bifurcation diagram for supercoiled DNA trefoil knots and a theorem to the effect that the minimum bending energy configuration a (2, q) torus knot shows self-contact along a circle. The theorem remains valid even when the integer q is even, i.e., when the "knot" reduces to a link of 2 unknots.

It is a macroscopic observation that knotted ropes or fishing lines break under tension at a knot site. Recently these observations were extended to molecular dimensions by demonstrating that knotted actin filaments break at the knot whereby molecular modeling studies indicated that this should be also the case for a knotted polyethylene chain. We investigate here the strength of different knots tied on the same cut of a fishing line or on cooked spaghetti. Among the simple knots we tested the overhand knot proved to be the weakest while the figure eight knot the strongest. Using high velocity camera and X-ray tomography, we localize sites where knotted strings break and compare it with curvature maps of tightly tied knots. Curvature and constriction are both needed to understand the rupture of a knottted polymer.

Magnetic knots are mathematical idealizations of knotted magnetic field distributions. In ideal magnetohydrodynamics conditions magnetic knots evolve according to a Lorentz force associated with the constituent lines of force and therefore experience a displacement and re-arrangement of their shape, while conserving their topology. This process is induced by the natural relaxation of the magnetic field towards minimum energy levels [2]. By using new equations for the Lorentz force [1], we show that in presence of inflexions (points where the curvature of the knot axis vanishes) magnetic knots become unstable and relax to inflexion-free braids [3]. This mechanism has important consequences for estimates of lower bounds of energy levels and find useful applications in the energetics of solar physics [4].
References:
[1] Ricca, R.L. 1997 Evolution and inflexional instability of twisted magnetic flux tubes. In "Solar Physics" vol. 172, 241-248.
[2] Ricca, R.L. 1998 Applications of knot theory in fluid mechanics. In "Knot Theory" (ed. V.F.R. Jones et al.), pp. 1-26. Banach Center Publs. vol. 42, Polish Academy of Sciences, Warsaw.
[3] Ricca, R.L. 1998, New developments in topological fluid mechanics: from Kelvin's vortex knots to magnetic knots. In "Ideal Knots" (ed. A. Stasiak et al.), pp. 255-273. Series on Knots and Everything vol. 19, World Scientific, Singapore.
[4] Ricca, R.L. & Berger, M.A. 1996, Topological ideas and fluid mechanics. "Phys. Today" vol. 49 (12), 24-30.

What, from the point of view of the knot theory, was the Gordian Knot? Was it a closed or open knot? As the legend says, nobody was able to untie it. Untie? As we know, any open knot can be untied - the algorithm to untie an open knot is known, one may lack patience, not skill, to untie any knot. On the other hand, had the Gordian Knot been a closed knot, no one would have chance to untie it, no matter what type of a knot it had been. As we know well, even the trefoil knot cannot be transformed into the trivial knot without cutting the rope. It could not be so simple. Most probably the problem of the Gordian Knot was totally different. As one of us (A.S.) suggested, it could have been the problem not of untying but disentangling a knot. The simplest problem one can pose here is: are there such a entangled conformations of the unknot tied on a finite piece of rope, which cannot be disentangled to the ground state torus conformation? This is a non-trivial problem. A firm answer to the question is not easy to find. As the numerical simulations we performed indicate, the answer is positive: there exist conformations of a the unknot, entangled in such a manner, that they cannot be disentangled to the torus conformation. We present an interesting, most probably simplest example.

A classic problem in science is the determination of the optimal arrangement of incompressible spheres in three dimensions in order to achieve the highest packing fraction. This simply stated problem has had a profound impact in many areas such as crystallization and melting of atomic systems. Here we propose and study the analogous problem of determining the optimal shapes of close-packed compact strings. The problem is a mathematical idealization of situations commonly encountered in biology, chemistry and physics of the optimal structure of folded polymeric chains. By using recent ideas developed for ideal representation of knots and by using stochastic optimization techniques we show that, when boundary effects are not dominant, helices with a special pitch-radius ratio are selected. Strikingly, the same special geometry is observed in helices in naturally occurring proteins.