Untangling knots via reaction-diffusion dynamics of vortex strings

Fabian Maucher†? and Paul Sutcliffe?

†Joint Quantum Centre (JQC) Durham-Newcastle, Department of Physics,Durham University, Durham DH1 3LE, United Kingdom.

?Department of Mathematical Sciences, Durham University, Durham DH1 3LE, United Kingdom.Email: fabian.maucher@durham.ac.uk, p.m.sutcliffe@durham.ac.uk

(Dated: April 2016)

We introduce and illustrate a new approach to the unknotting problem via the dynamicsof vortex strings in a nonlinear partial differential equation of reaction-diffusion type. Tountangle a given knot, a Biot-Savart construction is used to initialize the knot as a vortexstring in the FitzHugh-Nagumo equation. Remarkably, we find that the subsequent evolutionpreserves the topology of the knot and can untangle an unknot into a circle. Illustrative testcase examples are presented, including the untangling of a hard unknot known as the culprit.Our approach to the unknotting problem has two novel features, in that it applies field theoryrather than particle mechanics and uses reaction-diffusion dynamics in place of energy minimization.

PACS numbers: 47.32.cf, 02.10.Kn, 82.40.Ck, 87.19.Hh

The fundamental problem in knot theory is to de-termine whether a closed loop embedded in three-dimensional space is knotted. This unknotting prob-lem, and more generally the identification of knots,leads to deep mathematical connections throughoutthe natural sciences from quantum field theory [1]to the properties of DNA [2]. There has been con-siderable recent interest in knotted vortex strings infield theory in a range of areas from fluid dynam-ics [3] to ultracold atoms [4], with superfluid vortexstrings providing a canonical example. However, insuch systems there are reconnection events (wherecrossing strings exchange partners) which change theknot type and eventually all knots completely untie,accompanied by the production of a set of unknot-ted circles. A very recent comprehensive analysis ofthe untying pathways of initially knotted superfluidvortex strings suggests that some universal mecha-nisms are at work [5]. An exception to this classof field theories is the Skyrme-Faddeev model [6],where static minimal energy knotted vortex stringsexist [7], stabilized by a topological charge. How-ever, even in this system, reconnection events takeplace, therefore the knot type is not conserved dur-ing energy relaxation, and indeed an optimal mini-mal energy knot can appear via reconnection fromthe unknot.

The ubiquitous reconnection of vortex strings innonlinear physical fields has reduced their impact onknot theory. In the present paper we propose a novelapplication of vortex strings to knots by consider-ing a reaction-diffusion system in which the vortex

FIG. 1. The vortex string of the culprit unknot and thefield u in a planar slice.

strings appear to be averse to reconnection. In par-ticular, we present some illustrative test case exam-ples of unknots that untangle without reconnection,to yield a circular unknot. The dynamics of vortexstrings in this system is therefore in remarkable con-tradistinction to the universal properties discussedin [5]. Studying knots via topology preserving vor-tex string dynamics is a new paradigm connectingclassical field theory, knot theory and partial dif-ferential equations. In contrast to conventional knotuntangling methods, which apply particle mechanicsand energy minimization, our approach is novel in itsapplication of field theory and reaction-diffusion dy-namics to this problem. The use of reaction-diffusionsystems as a computation device is a current hottopic in the field of novel and emerging computing

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paradigms [8] and our approach to knot untanglingprovides a new application in this field.

The reaction-diffusion equation of interest in thispaper is the FitzHugh-Nagumo system, which is asimple mathematical model of cardiac tissue as anexcitable medium [9]. The nonlinear partial differ-ential equations are given by

∂u

∂t=

1

ε(u− 1

3u3− v) +∇2u,

∂v

∂t= ε(u+ β − γv),

(1)where u(r, t) and v(r, t) are the real-valued physicalfields defined throughout three-dimensional spacewith coordinate r, and where t denotes time. Theremaining variables are constant parameters that wefix to be ε = 0.3, β = 0.7, γ = 0.5 from now on.

In two-dimensional space this system, with the pa-rameter values given above, has rotating vortex so-lutions, often called spiral waves [9], with a periodT = 11.2 and u and v wavefronts in the form of aninvolute spiral with a wavelength λ = 21.3. Char-acteristic time and length scales are determined bythe parameters T and λ and in particular the vortexcore can be assigned a radius λ

2π . The centre of thevortex is the point at which |∇u×∇v| is maximal,and this quantity is localized in the vortex core.

Motivated by the observed short-range repulsiveforce between vortex cores, it was conjectured manyyears ago [10] that knotted vortex strings in thethree-dimensional system might preserve their topol-ogy under FitzHugh-Nagumo evolution. Numericalsupport for this conjecture was obtained [11] by sim-ulations on long time scales (thousands of vortex ro-tation periods) for the simplest knot and link (thetrefoil knot and the Hopf link). However, investi-gations of more complicated vortex string geome-tries or potential applications to knot theory haveremained elusive because of the difficulties in ob-taining more general initial conditions beyond sym-metric representations of simple knots.

In this paper, we remove this obstacle by adaptinga recent scheme [12], introduced for superfluid vor-tices, to the FitzHugh-Nagumo system. The methodinvolves a Biot-Savart construction and allows thecreation of initial conditions for a vortex string witharbitrary shape and topology. In detail, given anynon-intersecting closed curve K, one imagines thiscurve to be a wire carrying a constant current andcomputes the associated magnetic field B(r) usingthe Biot-Savart law

B(r) =1

2

∫K

(r− `)× d`|r− `|3

, (2)

where ` is a coordinate on K. The scalar potentialΦ(r), defined by B = ∇Φ is then computed by fixinga base point r0 and performing the line integral

Φ(r) =

∫C

B(r′) · dr′, (3)

where C is a curve that starts at r0 and ends atr whilst avoiding K. Finally, the initial FitzHugh-Nagumo fields u and v are obtained from the scalarpotential using the formulae

u = 2 cos Φ− 0.4, v = sin Φ− 0.4. (4)

The subtraction of the constants in (4) is motivatedby the fact that the centre of a vortex is roughly as-sociated with the field values (u, v) = (−0.4,−0.4).The scaling factor of 2 reflects the property that thefields outside the core of a single planar vortex per-form a periodic oscillation in the (u, v)-plane with arange in the u direction that is roughly twice that inthe v direction.

After a time scale of the order of a vortex rota-tion period T , the above construction generates avortex string along K. Fig. 1 displays an examplenumerical implementation of this construction in aregion of size 200×200×150 and after a time t = 60,where the FitzHugh-Nagumo equations are evolvednumerically using standard methods with Neumannboundary conditions. The vortex string is displayedby plotting the isosurface where |∇u ×∇v| = 0.1,and the same method is used to display all the vor-tex strings presented in this paper. The field u in aplanar slice is also displayed, to illustrate the spiralwavefronts that emanate from the vortex string.

Fig. 2 displays the knot already shown in Fig. 1,but more clearly as a projection without the u fieldin the (x, y)-plane. This knot has 10 crossings inthe given projection but is in fact an unknot knownas the culprit [13]. It is an example of a hard un-knot, which means that the number of crossingsmust first be increased in the process of untanglingthis unknot into a circle. The resulting evolutionis shown (to scale) in the remainder of Fig. 2 andis available as the movie file culprit.mpg in the sup-plementary material. This result demonstrates thatFitzHugh-Nagumo dynamics successfully untanglesthis unknot, preserving the topology whilst evolv-ing the geometry to the circular unknot. It is thefirst example, in any field theory, of vortex stringdynamics that untangles a knot and is free from re-connection. This field theory method seems at leastas powerful as the other three-dimensional untan-gling strategies, such as energy minimization, all of

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FIG. 2. The vortex string at t = 60, 1000, 2000, 3000, 4000, 4600, untangling the culprit unknot.

FIG. 3. A vortex string is displayed at the timest = 200, 500, 900, 1200, 1500, 1800, 1900, 2100, 2500, dur-ing the untangling of a 13-crossing unknot.

which seem significantly more powerful than combi-natorial untangling strategies, which are known tohave trouble with configurations like the culprit.

A second unknot example, with more initial cross-ings than the culprit, is displayed (to scale) in Fig. 3,using the same size grid. This figure illustrates theprocess in which this unknot with 13 initial crossingsis successfully untangled without reconnection. Theevolution is available as the movie unknot13.mpg inthe supplementary material.

As a final test of our new approach, we considerthe knot displayed in the first image in Fig. 4. The

initial knot has 7 crossings in this projection and, asevery gentleman will recognize, this is a knot usedto tie a cravat, if the two loose ends are then joined.The subsequent evolution is displayed (to scale) inthe remaining images in Fig. 4 and is available asthe movie cravat.mpg in the supplementary material.Again there are no reconnections and the dynamicssimplifies the geometry of the knot to reveal a trefoilknot with 3 crossings.

Vortex string reconnections are certainly not for-bidden in the FitzHugh-Nagumo model and we havebeen able to force reconnections by initializing avortex string with segments that are initially sep-arated by less than a core diameter, so that recon-nection takes place before the rotating spiral wavevortices have completely formed. However, the dy-namics appears to be averse to creating this situa-tion from fully formed vortices that have attainedseparations beyond a vortex core diameter (as illus-trated in Fig. 1). By a suitable choice of scale, anygiven knot can be initialized so that all segmentsare separated by more than a core diameter, as wehave done for the illustrative test case examples pre-sented in this paper. For these, and a number ofsimilar examples that we have investigated for a va-riety of initial knots, we find no evidence of stringreconnection, providing the initial knot has a mini-mal distance between segments that is greater thana vortex core diameter.

The examples we have studied so far are simpleenough that the absence of string collisions and re-connections can be verified manually, but to investi-gate the untangling of more complex examples, suchas the notoriously difficult Ochiai unknot [14], wouldrequire the implementation of some form of auto-matic string collision detection, to be certain thatthe knot topology is preserved. It could be that suf-ficiently complicated knots, beyond those studied so

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FIG. 4. The vortex string at t = 60, 400, 760, 1600, 2200, 2720, illustrating the simplification of a trefoil knot.

far, are not immune to vortex string reconnection,and it would be interesting to investigate this fur-ther and determine which, if any, types of knot arecapable of inducing this phenomenon.

Methods have been developed to reduce the fieldtheory dynamics to an effective dynamics of the vor-tex string [15, 16], but they apply only in the regimeof slight curvature and twist. It is therefore a chal-lenging open problem to develop an effective stringdynamics that is capable of reproducing the untan-gling motion described in the present paper and toexplain the aversion to string reconnection.

Simulated annealing methods have been devel-oped in mechanical models of knots [17] to re-solve the problem of the knot conformation becom-ing trapped in a local energy minimum. As ourapproach is based on reaction-diffusion evolution,rather than energy minimization, there is no issueregarding the knot conformation becoming trappedin a local energy minimum. The relevant questionfor reaction-diffusion evolution is whether the flowhas multiple attractors that can trap the knot con-formation or if there is only one unknotted attract-ing steady state. This is an interesting mathemati-cal problem that requires further study to determinewhether this approach is an improvement on en-ergy minimization methods, with regard to trappedconformations. As untangling via reaction-diffusionevolution appears to explore very different untan-gling pathways and knot geometries to energy mini-mization, this may lead to some new understandingof the properties of physical knots, such as thosefound in DNA, which may also utilize chemical orother messengers to exchange information betweendifferent arcs of the knot, as modelled by wave in-teractions in reaction-diffusion dynamics.

By adapting a new construction of vortex stringinitial conditions to the FitzHugh-Nagumo equa-tion, we have been able to investigate the reaction-

diffusion evolution of complex vortex string geome-tries with arbitrary topology. Remarkably, in con-trast to other field theories, we find that the vortexstrings preserve their knot topology and evolve with-out reconnection to untangle a knot into a simplifiedgeometrical form. The fact that very complicatedinitial geometries untangle without reconnection isunexpected and generates a new open problem: toexplain how reaction-diffusion dynamics, which hasno conserved quantities, is able to preserve the knottopology whilst simultaneously simplifying the ge-ometry. The implications of an understanding of thisbehaviour could have relevance in a range of topicsin physics, chemistry, biology and mathematics.

The theory of gradient flows for energy function-als on knots has various significant technical chal-lenges even to simply construct equations for theflow and to prove that short time solutions exist(see for example the work [18] on the gradient flowfor the Mobius energy). A significant advantage ofthe reaction-diffusion flow of vortex strings is theabsence of all these difficulties.

Finally, there are several chemical and biologicalsystems supporting spiral wave vortices, from theoscillating Belousov-Zhabotinsky redox reaction tothe chemotaxis of slime mould. All are describedby similar reaction-diffusion equations that possessvortex strings, so it is not beyond the realms ofpossibility that in the future a chemical or biologicalsystem might be engineered that could untangleknots.

ACKNOWLEDGEMENTS

We thank Hayder Salman and Lauren Scanlon foruseful discussions. This work is funded by the Lev-erhulme Trust Research Programme Grant RP2013-K-009, SPOCK: Scientific Properties Of ComplexKnots, and the STFC grant ST/J000426/1.

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[1] E. Witten, Commun. Math. Phys. 121, 351 (1989).[2] S.A. Wasserman and N.R. Cozzarelli, Science 232,

951 (1986).[3] D. Kleckner and W.T.M. Irvine, Nat. Phys. 9, 253

(2013).[4] D.S. Hall, M.W. Ray, K. Tiurev, E. Ruokokoski,

A.H. Gheorghe and M. Mottonen, Nat. Phys.(2016), doi:10.1038/nphys3624.

[5] D. Kleckner, L.H. Kauffman and W.T.M. Irvine,How superfluid vortex knots untie, arXiv:1507.07579

[6] L. Faddeev and A.J. Niemi, Nature 387, 58 (1997).[7] R.A. Battye and P.M. Sutcliffe, Phys. Rev. Lett. 81,

4798 (1998).[8] A. Adamatzky, B. De Lacy Costello and T. Asai,

Reaction-Diffusion Computers, Elsevier, 2005.[9] A.T. Winfree, The Geometry of Biological Time,

Springer-Verlag, 2001.[10] A.T. Winfree and S.H. Strogatz, Nature 311, 611

(1984).[11] P.M. Sutcliffe and A.T. Winfree, Phys. Rev. E 68,

016218 (2003).[12] H. Salman, Phys. Rev. Lett. 111, 165301 (2013).[13] L.H. Kauffman and S. Lambropoulou, Ser. Knots

Everything 46, 187 (2012).[14] M. Ochiai, S.M.F. Asterisque 192, 7 (1990).[15] J.P. Keener, Physica D 31, 269 (1988).[16] V.N. Biktashev, A.V. Holden and H. Zang, Philos.

Trans. R. Soc. London A347, 611 (1994).[17] M. Huang, R.P. Grzeszczuk and L.H. Kauffman, in

IEEE Visualization 279 (1996).[18] Z.X. He, Comm. Pure Appl. Math. 53, 399 (2000).

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