Confirmed Speakers with their titles


1 Mitchell Berger (UK)
``Magnetic Writhe and Braiding''

Mathematical methods arising from the analysis of magnetic fields may have relevance in other areas of research. First I will discuss definitions of writhe for curves with endpoints. Writhe is a measure of coiling traditionally defined for closed curves.

Kinked (buckled) magnetic flux tubes in the solar atmosphere clearly exhibit writhe, but have endpoints on the solar surface. Fortunately, simple definitions for open writhe arise naturally from the requirement of consistency with magnetic helicity integrals (analogous to Gauss linking integrals). These definitions can be applied to any open curves.

Secondly I will discuss braiding of magnetic field lines in solar x-ray loops. Turbulent motions at the surface of the sun randomly move the field line endpoints about each other, braiding the lines above. This braiding is removed by reconnection events, resulting in small solar flares. Balancing random input of braiding with not-so-random reconnection may lead to a self-organized braid structure with power law structural properties.



2 Jason Cantarella (USA)
"A new proof that every tight knotted tube (other than the unknot) has a self-contact"

Abstract: In this talk we use the theory of ropelength criticality to completely classify tubes which are critical for minimizing length but have no self contacts. We first show that the centerline of such a tube has constant curvature and satisfies a certain differential equation. Then we observe that the differential equation implies that each curve has a conserved vector quantity. We can then classify the solutions of the equations as circles (where the conserved vector is the normal to the plane of the circle), helices (where the conserved vector is the axis of the helix) or more complicated "iterated helices" (where the conserved vector is again a sort of axis).
Analyzing the conserved vector shows that the only such curve that closes is the circle, thus reproving a recent result of Durumeric in a different way. This talk is joint work with Joe Fu (UGA), Rob Kusner (UMass), John Sullivan (TU Berlin) and Nancy Wrinkle (NEIU).


3 Yuanan Diao (USA)
``Jones Polynomials of Knots Formed by Repeated Tangle Replacement Operations''


Abstract: We prove that the Jones polynomial of a link diagram obtained through repeated tangle replacement operations can be computed by a sequence of suitable variable substitutions in simpler polynomials. For the case that all the tangles involved in the construction of the link diagram have at most $k$ crossings (where $k$ is a constant independent of the total number $n$ of crossings in the link diagram), we show that the computation time needed to calculate the Jones polynomial of the link diagram is bounded above by $O(n^k)$. In particular, we show that the Jones polynomial of any Conway algebraic link diagram with $n$ crossings can be computed in $O(n^2)$ time. A consequence of this result is that the Jones polynomial of any Montesinos link and two bridge knot or link of $n$ crossing can be computed in $O(n^2)$ time.

This is a joint work with Claus Ernst and Uta Ziegler.


4 Giovanni Dietler (Switzerland)
"Single Molecule Imaging Study of the Effects of Topology on the Statistical Properties of DNA"

5 Klaus Ernst (USA)
``The Linearity of the Ropelengths of Conway Algebraic Knots in Terms of Their Crossing Numbers''

Abstract: For a knot or link $K$, let $L(K)$ denote the ropelength of $K$ and let $Cr(K)$ denote the crossing number of $K$. An important problem in geometric knot theory concerns the relationship between $L(K)$ and $Cr(K)$ (or intuitively, the relationship between the amount of a rope needed to tie a particular knot and the complexity of the knot). We show that there exists a constant $a>0$ such that for any knot $K$ that allows a special knot diagram $D$ (called Conway algebraic knot diagram) with $n$ crossings, then $L(K)\le a\cdot n$. Furthermore, if $D$ is alternating (but not necessarily reduced and in fact it may not have a minimal alternating diagram that is algebraic), then $L(K)\le a\cdot Cr(K)$. The approach used here can be applied to a larger class of knots, namely those formed by repeated tangle replacements. Interestingly, it has been shown by the same authors that the Jones polynomials of these knots can be computed in polynomial time.


6 Tom Kephart (USA)
"Knots and Links in Semi-classical Systems"

Only the simplest linking has been studied thoroughly in physical systems. Higher order linking has yet to be exploited. We consider systems with knotted and higher linked components. Examples range from bio-systems to quantum chromodynamics.




7 Stephen Levene (USA)
TBA

8 Cristian Micheletti (Italy) SISSA
"Simulations of knotting in confined circular DNA"


ABSTRACT. The packing of DNA inside bacteriophages arguably yields the simplest example of genome organisation in living organisms. As an assay of packing geometry, the DNA knot spectrum produced upon release of viral DNA from the P4 phage capsid has been analyzed, and compared to results of simulation of knots in confined volumes. We present new results from extensive stochastic sampling of confined self-avoiding and semi-flexible circular chains with volume exclusion. The physical parameters of the chains (contour length, cross section and bending rigidity) have been set to match those of P4 bacteriophage DNA. By using advanced sampling techniques, involving multiple Markov chain pressure-driven confinement combined with a thermodynamic reweighting technique, we establish the knot spectrum of the circular chains for increasing confinement up to the highest densities for which available algorithms can exactly classify the knots. Compactified configurations have enclosing hull diameter about 2.5 times larger that the P4 calliper size. The results are discussed in relation to the recent experiments on DNA knotting inside the capsid of a P4 tailless mutant. Our investigation indicates that confinement favours chiral knots over achiral ones, as found in the experiments. However, no significant bias of torus over twist knots is found, contrary to the P4 results. The result poses a crucial question for future studies of DNA packaging in P4: is the discrepancy due to the insufficient confinement of the equilibrium simulation or does it indicate that out-of-equilibrium mechanisms (such as rotation by packaging motors) affect the genome organization, hence its knot spectrum in P4 ?



9 Olivier Pierre-Louis (UK)
``Knots in stiff strings''

We report on the geometry and mechanics of knotted strings. We focus on the situation where the string is stiff (it has a large bending rigidity), and thin (its width is much smaller than its length).

We find that: (i) the equilibrium energy depends on the type of knot as the square of the bridge number; (ii) braid localization is a general feature of stiff strings entanglements; (iii) there is an upper bound for the multiplicity of the braids and contact points in the ground state. (iv) Finally, a general confinement inequality is used to derive an upper bound on the knot gyration radius.

We shall also discuss the asymptotic behavior of the knot when the filament width is small, both in the presence and in the absence of torsion (twist) energy. We conjecture a universal ground state geometry for thin strings with torsion rigidity in the presence of a large twists.

Ref: R. Gallotti, O. Pierre-Louis, Phys. Rev. E 75, 031801 (2007).


10 Eric Rawdon (USA)
``The Effect of Knotting on the Size and Shape of Polymers''

We use numerical simulations to investigate how the chain length and topology of freely fluctuating knotted polymer rings affect their size and shape. In particular, we analyze different types of geometric containers that envelope polymer configurations and describe the similarities and differences between them. This work has been done in collaborations with Akos Dobay, John Kern, Kenneth Millett, Michael Piatek, Patrick Plunkett, and Andrzej Stasiak.




11 Chris Soteros (Canada)
``THE LINKING PROBABILITY FOR 2-COMPONENT LINKS WHICH SPAN A LATTICE TUBE''

Sumners and Whittington (1988) investigated questions about knottedness of a closed curve embedded in the three dimensional integer lattice $\mathbb{Z}^3$ (i.e. self-avoiding polygons on the simple cubic lattice).

They proved that all but exponentially few sufficiently long self-avoiding polygons are knotted. Using a self-avoiding polygon to model a ring polymer, this proved the long standing Frisch-Wasserman-Delbruck conjecture that the probability for a ring polymer to be knotted goes to one as the polymer size increases. One can ask a similar question about the probability that two ring polymers are linked, and in particular, under what circumstances does this probability go to one as the size of the polymers increases. To address this, we consider two self-avoiding polygons

(2SAPs) each of which spans a tubular sublattice of $\mathbb{Z}^3$. A pattern theorem is proved for 2SAPs, that is any proper pattern (a local configuration in the middle of a 2SAP) occurs in all but exponentially few sufficiently large 2SAPs. C. Ernst established that there exists a pattern which, should it occur in a 2SAP, guarantees that the 2SAP is topologically linked. Thus the pattern theorem then establishes that all but exponentially few sufficiently large 2SAPs are topologically linked. Moreover we also use the pattern theorem and a ``coin-tossing argument'' to prove that the linking number $Lk$ of an $n$ edge 2SAP $G_n$ satisfies $\lim_{n\rightarrow\infty}\mathbb{P}\big(|Lk(G_n)|\geq f(n)\big)=1$ for any function $f(n)=o(\sqrt{n})$. Hence the probability of a non zero linking number for a 2SAP approaches one as the size of the 2SAP goes to infinity. It is also established, due to the tube constraint, that the linking number of an $n$ edge 2SAP grows at most linearly in $n$.

This work was done in collaboration with M. Atapour, C. Ernst and S. Whittington.



12 Andrzej Stasiak (Switzerland)
``DNA supercoiling inhibits DNA knotting''

Yannis Burnier, Julien Dorier & Andrzej Stasiak


Despite the fact that in living cells DNA molecules are long and highly crowded, they are rarely knotted. DNA knotting interferes with the normal functioning of the DNA and, therefore, molecular mechanisms evolved that maintain the knotting and catenation level below that which would be achieved if the DNA segments could pass randomly through each other. Biochemical experiments with torsionally relaxed DNA demonstrated earlier that type II DNA topoisomerases that permit inter- and intramolecular passages between segments of DNA molecules use the energy of ATP hydrolysis to select passages that lead to unknotting rather than to the formation of knots. Using numerical simulations, we identify here another mechanism by which topoisomerases can keep the knotting level low. We observe that DNA supercoiling, such as found in bacterial cells, creates a situation where intramolecular passages leading to knotting are opposed by the free energy change connected to transitions from unknotted to knotted circular DNA molecules.


13 Attilio Stella (Italy)
``Topological signatures of the thermodynamics of swollen and globular polymer rings''

A. L. Stella- Dipartimento di Fisica "Galileo Galilei" and Sezione INFN, Universita' di Padova, Italy.

The identification of the topological invariants affecting the thermodynamics of a long, closed chain with a knot is a central problem in polymer statistics. In the swollen regime the number of prime components is expected to be the essential topological property entering in the asymptotic corrections to the entropy. However, recent results by our group show that the topology of each component determines specific extra corrections, giving the possibility of distinguishing between different knots. The relevant invariant involved appears to be the ideal knot length. In the globular phase, below the theta temperature, knots are expected to be delocalized along the whole chain. At first sight, this situation seems to reduce, with respect to the swollen case, the possibility for topology of affecting corrections to the asymptotic free energy, since prime knot components are not localized in this case. Extensive simulations of interacting self-avoiding polygons on the cubic lattice have led to establish a power law behavior for the rank-ordered probability of different topologies as a function of order and to identify a key correction term for the free energy depending on the minimal crossing number of the knot. This indicates that this number is the relevant topological invariant in the globular phase. We will review the results of several numerical experiments in which the role played by this invariant is clearly manifested. In these experiments suitable constraints imposed to the configurations of the chain allow to establish laws governing the topologically delocalized globular state and to verify remarkable effects occurring,e.g., in the translocation of a ring polymer through a pore.

M. Baiesi, E. Orlandini, A.L. Stella, PRL 99, 058301 (2007).
M. Baiesi, E. Orlandini, A. L. Stella, F. Zonta, paper in preparation.



14 Joanna I. Sulkowska (USA)
TBA

15 Peter Virnau (Germany)
``Knots in globular polymers and proteins''

In this talk, I will discuss the occurrence of knots in numerical simulations of a model polyethylene, spanning high temperature (coil) and low temperature (globule) phases and investigate the influence of polymer stiffness in confined geometries. I will also present statistics on the location and the distribution of knot sizes in random walks. Although globular homopolymers display an abundance of knots, only about one in a hundred protein structures are knotted. I will present an overview of knotted proteins from the current version of the Protein Data Bank. Among others, I will present the most intricate protein knot known to date and and explain how knots probably appeared in the cause of evolution.




16 Stu G. Whittington (Canada)
"Counting almost unknotted embeddings of graphs and surfaces"

17 Julia Yeomans (UK)
``Polymer dynamics in confined spaces''

The Rudolf Peierls Centre for Theoretical Physics,
1 Keble Road, Oxford, OX1 3NP, UK


Recent advances in mesoscale simulation techniques [1] have allowed increased scope to investigate polymer hydrodynamics. In this talk I shall describe recent work at Oxford on two problems that concern polymers moving in confined geometries.

Firstly we consider drops in a binary fluid, which stretch and break up in a shear flow. We show that adding polymers can either suppress or enhance drop break-up, for different regimes of capillary number.

Secondly, we return to work of Daoud and De Gennes [2] who used a blob picture to argue that the probability that a polymer can be pushed into a narrow channel by a flow field is determined by the temperature and fluid viscosity, but is independent of the length of the polymer chain. We use lattice Boltzmann and stochastic rotation dynamics simulations to test these predictions.


[1] P. Ahlrichs and B. Dunweg, J. Chem. Phys. 111 8225 (1999).
[2] M Daoud and P.G. de Gennes, J. Phys. France 38 85 (1977)




18 Todd O Yeates (USA)
TBA

19 Lynn Zechiedrich (USA)
TBA