Bulletin of the American Physical Society
APS March Meeting 2017
Volume 62, Number 4
Monday–Friday, March 13–17, 2017; New Orleans, Louisiana
Session K17: Knotting in Filaments and FieldsFocus

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Sponsoring Units: GSOFT Chair: Mark Dennis, University of Bristol Room: 276 
Wednesday, March 15, 2017 8:00AM  8:36AM 
K17.00001: The inevitability of knotting: Polymers, filaments and surfaces Invited Speaker: Stu Whittington Knots are ubiquitous in physics and in biology. They occur in biopolymers such as DNA and proteins, in vortices in fluids, in optical beams, in liquid crystals and in surfaces. Indeed, long flexible objects are knotted with high probability. This talk will review some rigorous results about the inevitability of knotting in ring polymers and in surfaces, and will discuss the extension of these results to other physical systems. For many lattice models one can prove that knotting is a high probability event, by showing that a local structure that ensures knotting occurs with high probability somewhere in large flexible structures. [Preview Abstract] 
Wednesday, March 15, 2017 8:36AM  8:48AM 
K17.00002: Asymptotic higher linking in volume preserving flows Rafal Komendarczyk The topology of orbits in volumepreserving flows is of special interest in the area of fluid dynamics and plasma physics. The wellknown helicity invariant measures an asymptotic average linking number of orbits of the flow and gives an estimate for the $L^2$energy of the field. Invariants of flows obtained from higher linking numbers were only derived under special assumptions on the domain of the field or rely on the special features of the vector field. In this talk, I will present preliminary results on an irregular asymptotic invariant obtained from the triple linking number. [Preview Abstract] 
Wednesday, March 15, 2017 8:48AM  9:00AM 
K17.00003: Virtual knotting in proteins and other open curves Keith Alexander, Alexander Taylor, Mark Dennis Long filaments naturally knot, from string to longchain molecules. Knotting in a filament affects its properties, and may be very stable or disappear under slight manipulation. Knotting has been identified in protein backbones for which these mechanical constraints are of fundamental importance to their function [1], although they are open curves in which knots are not mathematically well defined; knotting can only be identified by closing the ends of the chain. We introduce a new method for resolving knotting in open curves using virtual knots [2], a wider class of topological objects that do not use a classical closure, capturing the topological ambiguity of open curves. Having analysed all proteins in the Protein Data Bank by this new scheme, we recover and extend previous knotting results, and identify topological interest in some new cases. The statistics of virtual knots in proteins are compared with those of Hamiltonian subchains on cubic lattices [3], identifying a regime of open curves in which the virtual knotting description is likely to be important [4].\\{}[1] M Jamroz et al, Nuc Acids Res 43, D306–14 (2014)\\{}[2] L H Kauffman, Eur J Combin 20, 663–90 (1999)\\{}[3] R C Lua & A Y Grosberg, PLoS Comput Biol 2, e45 (2006)\\{}[4] K Alexander et al, submitted (2016) [Preview Abstract] 
Wednesday, March 15, 2017 9:00AM  9:12AM 
K17.00004: Characterizing knotting for polymers in tubes and nanochannels Christine Soteros, Nicholas Beaton, Jeremy Eng Motivated in part by experimental studies of DNA in viral capsids or in nanochannels, there is interest in understanding and characterizing the entanglement complexity of confined polymers. For this, one quantity of interest has been the ``size" of the ``knotted part" of a polymer. With such a measure, one can characterize knotting as ``local", when the size of the knotted part is small compared to the total length of the polymer, or otherwise ``nonlocal". One size measure is to associate knottypes to subarcs of a knotted polymer chain and use the arclength of the smallest knotted subarc as the knotsize. Using this, we study lattice models of polymers to explore the effects of confinement to a lattice tube on the likelihood of nonlocal versus local knotting. We classify knotted patterns as either nonlocal or local, depending on whether they can occur in a polygon in a nonlocal way or not. For an equilibrium model of ring polymers in a tube subject to a tensile force $f$, we prove results about the likelihood of occurrence of the two types of knotted patterns as a function of polymer length and provide evidence that nonlocal knot configurations are more likely than local ones, regardless of the strength or direction (stretching or compressing) of the force $f$. [Preview Abstract] 
Wednesday, March 15, 2017 9:12AM  9:24AM 
K17.00005: Pore translocation of knotted DNA chains Antonio Suma, Cristian Micheletti Biopolymers, such as DNA, can be long enough to become spontaneously knotted. This can have detrimental effects on their functionality in biological contexts, and in singlemolecule manipulation experiments too. A relevant example is the translocation of DNA through biological or solidstate nanopores, which can become hindered by the presence of knots. We report here on a first systematic theoretical and computational investigation of such translocation for knotted DNA chains and elucidate the sophisticated, and even counterintuitive interplay of DNA topology, geometry and the strength of the applied tractive force~\footnote{ A. Rosa, M. Di Ventra and C. Micheletti. \textit{Phys. Rev. Lett}, 2012, 109 , 118301}~\footnote{ A.Suma, A. Rosa and C. Micheletti. \textit{ACS Macro Letters}, 2015, 4(12), 14201424}. [Preview Abstract] 
Wednesday, March 15, 2017 9:24AM  9:36AM 
K17.00006: Tying Knots in DNA with Holographic Optical Tweezers Mervyn Miles, David Foster, Annela Seddon, David Phillips, David Carberry, Miles Padgett, Mark Dennis 
Wednesday, March 15, 2017 9:36AM  9:48AM 
K17.00007: Abstract Withdrawn Knots have been found in many proteins. The problem is to explain how a knot forms. For this we develop a Landau model that describes protein folding in terms of soliton formation along a one dimensional chain. When the ambient temperature increases, a folded protein unfolds and assumes the geometric character of a selfavoiding random walk. When ambient temperature then decreases, the protein folds back to its original space filling conformation. If there is a knot along the protein, we show how increase in ambient temperature opens it, and when the ambient temperature decreases we show how the protein becomes again knotted. However, we find that there is a critical temperature: If the ambient temperature exceeds this critical value the protein is no longer able to form a knot when the temperature decreases. We show how this kind of phase transition takes place in an actual knotted protein. 
Wednesday, March 15, 2017 9:48AM  10:00AM 
K17.00008: Optical vortex knots in tightlyfocused light beams Mark Dennis, Danica Sugic Optical vortices, that is, zero lines of complex amplitude in a propagating light field, can be knotted or linked in a controlled way [1]. This was demonstrated previously in experiments where a computercontrolled hologram determined the amplitude of paraxial laser light [1], meaning the longitudinal extent of the knot was several orders of magnitude larger than its width. We describe what happens to these optical knots when the transverse width of the beam, and hence the knot, is reduced. Outside the paraxial regime, the field's polarization becomes highly inhomogeneous, and knotted structures occur in a variety of polarization singularities [2]. We propose experiments realising these knotted polarization structures in tightlyfocused beams, which should yield optical knots of unit aspect ratio, of several optical wavelengths in size, which could be suitable for embedding knotted defect structures in liquid crystals, BoseEinstein condensates [3] and photopolymers. [1] M R Dennis et al, Nature Physics 6, 117129 (2010); [2] J F Nye and J Haljnal, Proc R Soc A 409, 2136 (1987); [3] F Maucher et al, arXiv:1512.01012 (2015). [Preview Abstract] 
Wednesday, March 15, 2017 10:00AM  10:12AM 
K17.00009: Kahler Structures on Spaces of Framed Curves Tom Needham Moduli spaces of loops in Euclidean space have been studied from a variety of perspectives due to their applications to fluid dynamics, statistical physics of polymers and shape recognition. We consider the moduli space of Euclidean similarity classes of parameterized framed loops. This space is an infinitedimensional Kahler manifold; in fact we show that it is isomorphic to an infinitedimensional Grassmannian with a natural Kahler structure. This result gives connections between previous results on loop spaces by YounesMichorShahMumford (in the context of image recognition) and MillsonZombro (in the context of symplectic geometry). The moduli space has many interesting Hamiltonian group actions, such as the diffeomorphism group of the circle acting by reparameterizations. The geodesic distance between the orbits of this group can be computed very efficiently, giving a new algorithm for shape recognition of ring polymers and oriented trajectories. [Preview Abstract] 
Wednesday, March 15, 2017 10:12AM  10:24AM 
K17.00010: Knots in light that persist with time Hridesh Kedia, Daniel PeraltaSalas, William T.M. Irvine Knots in the vortex lines of an inviscid (ideal) fluid persist forever. This led Lord Kelvin to hypothesize that atoms were vortex knots in the ideal aether, starting the field of mathematical knot theory. Surprisingly, knots in the lines of the magnetic (electric) field were shown to persist forever in recently discovered solutions to Maxwell's equations. However, in general, knots in the lines of the magnetic (electric) field do not persist. A natural question arises: when do knots in the lines of the magnetic (electric) field persist? We address this question with the aim of designing knotted light fields, by exploring connections between Maxwell's equations and ideal fluid flow. [Preview Abstract] 
Wednesday, March 15, 2017 10:24AM  10:36AM 
K17.00011: A knotted complex scalar field for any knot Benjamin Bode, Mark Dennis Threedimensional field configurations where a privileged defect line is knotted or linked have experienced an upsurge in interest, with examples including fluid mechanics [1], quantum wavefunctions, optics [2], liquid crystals [3] and skyrmions [4]. We describe a constructive algorithm to write down complex scalar functions of threedimensional real space with knotted nodal lines, using trigonometric parametrizations of braids. The construction is most natural for the family of lemniscate knots which generalizes the torus knot and figure8 knot [5], but generalizes to any knot or link. The specific forms of these functions allow various topological quantities associated with the field to be chosen, such as the helicity of a knotted flow field. We will describe some applications to physical systems such as those listed above. [1] H Moffatt, J Fluid Mech 35, 117129 (1969) [2] M R Dennis et al, Nature Physics 6, 118121 (2010) [3] T Machon and G Alexander, Phys Rev Lett 113, 027801 (2014) [4] P Sutcliffe, Proc R Soc A 463, 30013020 (2007) [5] B Bode, M R Dennis et al, arXiv:1611.02563 (2016) [Preview Abstract] 
Wednesday, March 15, 2017 10:36AM  10:48AM 
K17.00012: Stability of knotted vortices in wave chaos Alexander Taylor, Mark Dennis Large scale tangles of disordered filaments occur in many diverse physical systems, from turbulent superfluids to optical volume speckle to liquid crystal phases. They can exhibit particular large scale random statistics despite very different local physics. We have previously used the topological statistics of knotting and linking to characterise the large scale tangling, using the vortices of threedimensional wave chaos as a universal model system whose physical lengthscales are set only by the wavelength [1]. Unlike geometrical quantities, the statistics of knotting depend strongly on the physical system and boundary conditions. Although knotting patterns characterise different systems, the topology of vortices is highly unstable to perturbation, under which they may reconnect with one another. In systems of constructed knots, these reconnections generally rapidly destroy the knot, but for vortex tangles the topological statistics must be stable. Using large scale simulations of chaotic eigenfunctions, we numerically investigate the prevalence and impact of reconnection events, and their effect on the topology of the tangle. [1] A J Taylor and M R Dennis, Nat Comm 7, 12346 (2016) [Preview Abstract] 
Wednesday, March 15, 2017 10:48AM  11:00AM 
K17.00013: Conservation of Writhe Helicity Under AntiParallel Reconnection De Witt Sumners Reconnection is a fundamental event in many areas of science, from the interaction of vortices in classical and quantum fluids, and magnetic flux tubes in magnetohydrodynamics and plasma physics, to sitespecific recombination in DNA. The helicity of a collection of flux tubes can be calculated in terms of writhe, twist and linking among tubes. We prove that the writhe helicity is conserved under antiparallel reconnection [1]. We will discuss the mathematical similarities between reconnection events in biology and physics, and the relationship between iterated reconnection and curve topology. We will discuss helicity and reonnection in a tangle of confined vortex circles in a superfluid. Support from a Simons Foundation Collaboration Grant for Mathematicians is gratefully acknowledged. [1] Laing C.E., Ricca R.L. {\&} Sumners D.W. (2015) Conservation of writhe helicity under antiparallel reconnection, Nature Scientific Reports 5:9224 \textbar DOI: 10.1038/srep09224. [Preview Abstract] 
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