Knots and Physics

Knots and physics http://www-groups.dcs.st-and.ac.uk/~history/PrintHT/Knots_and_phys...

1 de 5 30/4/2007 16:18

History topic: Topology and Scottish mathematical physics

The Scottish mathematical physicists referred to in the title are Thomson, Maxwell and Tait. These threebecame involved in topological concepts, in particular knot theory, because it entered their physicalconsiderations in a natural way. In 1847 Listing published Vorstudien zur Topologie and then Riemannpublished important papers on complex analysis in 1851 and 1857 which investigated connectivity andRiemann surfaces. The Scottish mathematical physicists did not know of these papers until much later butHelmholtz, whose paper of 1858 directly influenced them, built much on Riemann's ideas.

In 1858 Helmholtz published his important paper in Crelle's Journal on the motion of a perfect fluid.Helmholtz's paper Uber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungenentsprechen began by decomposing the motion of a perfect fluid into translation, rotation anddeformation. It was this aspect which first interested Tait who saw that by using Hamilton's quaternionshe could express the fluid velocity as a "vector function". However the ideas in the paper whicheventually led the Scottish mathematical physicists to topological considerations concerned vortex linesand vortex tubes. Helmholtz defined vortex lines as lines coinciding with the local direction of the axis ofrotation of the fluid, and vortex tubes as bundles of vortex lines through an infinitesimal element of area.Helmholtz showed that the vortex tubes had to close up and also that the particles in a vortex tube at anygiven instant would remain in the tube indefinitely so no matter how much the tube was distorted it wouldretain its shape.

Helmholtz was aware of the topological ideas in his paper, particularly the fact that the region outside avortex tube was multiply connected which led him to consider many-valued potential functions. Hedescribed his theoretical conclusions regarding two circular vortex rings with a common axis ofsymmetry in the following way:-

If they both have the same direction of rotation they will proceed in the same sense, and thering in front will enlarge itself and move slower, while the second one will shrink and movefaster, if the velocities of translation are not too different, the second will finally reach thefirst and pass through it. Then the same game will be repeated with the other ring, so the ringwill pass alternately one through the other.

As we have mentioned Tait's first interest in Helmholtz' paper was because he saw applications ofquaternions there. It was not until 1867 that Tait verified Helmholtz' theoretical claims regarding twocircular vortex rings with experiments with smoke rings. He used two boxes each with a rubberdiaphragm which shot out white smoke rings when the diaphragm was struck. Thomson wrote toHelmholtz on 22 January 1867:-

... a few days ago Tait showed me in Edinburgh a magnificent way of producing [vortexrings]. We sometimes can make one ring shoot through another, illustrating perfectly yourdescription; when one ring passes near another, each is much disturbed, and is seen to be ina state of violent vibration for a few seconds, till it settles again into its circular form. ... Thevibrations make a beautiful subject for mathematical work.

These experiments were to have a major influence on Thomson who saw the permanence of form as apossible explanation for atoms and therefore explain the way that the different elements could be built. Itis remarkable that Thomson was able to develop his ideas quickly enough that he could publish On vortexatoms in the Proceedings of the Royal Society of Edinburgh still in 1867. In this paper he wrote:-

Leucretius's atom does not explain any of the properties of matter without attributing them to


2 de 5 30/4/2007 16:18

the atom itself ... The possibility of founding a theory of elastic solids and liquids on thedynamics of closely packed vortex atoms may be reasonably anticipated.

Although was now know that Thomson was completely wrong, there is much in the way of correctreasoning in the quote we have just given. It was an idea which also led to interesting mathematicalquestions. As he wrote in the same paper:-

A full mathematical investigation of the mutual action between two vortex rings of any givenvelocities passing one another in any two lines, so directed that they never come nearer toone one another than a large multiple of the diameter of either, is a perfect mathematicalproblem; and the novelty of the circumstances contemplated presents difficulties of anexciting character. Its solution will be the proposed new kinetic theory of gases.

Thomson, as most scientists of his time, viewed space as being filled with a perfect fluid, the aether. Thevortex atoms were then knotted tubes of aether which, by Helmholtz's theory, retained their form despitebeing distorted. This stability of vortices explained the stability of atoms. The vortex atoms, being builtfrom the aether, required no special material. The different elements were accounted for by atomscomposed of different knots or links and oscillations of the knots would, Thomson believed, explain thespectral lines which were characteristic of the different elements.

Tait began to think about knots and Thomson's second paper on vortex atoms, which appeared in 1869,included diagram of knots and links drawn by Tait. Long before this, however, Maxwell had entered thediscussions which went on in letters exchanged by the three Scottish mathematical physicists. He wasinterested in knots because of electromagnetic considerations and in a letter to Tait written on the 4December 1867 he rediscovered an integral formula counting the linking number of two closed curveswhich Gauss had discovered, but had not published, in 1833. Maxwell also gave equations in threedimensions which represented knotted curves.

However just when their work was proceeding rapidly, a major problem arose. Bertrand claimed thatHelmholtz's 1858 paper, on which the idea of vortex atoms was based, was wrong. Maxwell was alreadyintrigued by the problems and was not convinced that Bertrand's objections dealt a serious blow toHelmholtz's paper. In September 1868 Maxwell wrote several manuscripts which study knots and links.He set out the basic problem of the classification of knots and links as follows:-

Let any system of closed curves in space be given and let them be supposed capable of havingtheir forms changed in any continuous manner, provided that no two curves or branches of acurve ever pass through the same point of space, we propose to investigate the necessaryrelations between the positions of the curves and the degree of complication of the differentcurves of the system.

Maxwell considered two-dimensional projections of links and devised a way of coding the diagrams toindicate which curve was above and which below at crossings on the projections.

He then looked at ways of modifying the diagrams without changing the link or knot. For a regionbounded by one arc Maxwell noted that the region could be eliminated by uncoiling the curve. Forregions bounded by two arcs, he noted that there were two cases, one where the arcs could be separatedand the region eliminated, the other where this could not be done without making changes in other partsof the diagram. For regions bounded by three arcs Maxwell noted that again there were two cases:-

In the first case any one curve can be moved past the intersection of the other two withoutdisturbing them. In the second case this cannot be done and the intersection of two curves isa bar to the motion of the third in that direction.

Although his approach contained no mathematical rigour, still it is interesting to note that at this earlystage Maxwell had defined the "Reidemeister moves" which would be shown to be the fundamentalmoves in modifying knots in the 1920s.


3 de 5 30/4/2007 16:18

In a a second manuscript Maxwell considered a region of space bounded by one external surface of genusn and m internal surfaces of genus n1, n, ..., nm and showed that the region possessed N = n+n1+n+ ...+nm cycles and was (N+1)-ly connected. Now in modern terminology Maxwell was claiming that the firstBetti number of the region was N. Again we should note that Maxwell did not give precise mathematicaldefinitions of the concepts he was dealing with so no rigorous proof was possible. It is reasonable to askhow he then found the correct answer. The reason was that Maxwell, and for that matter Thomson too,reached their correct results using correct physical understanding, rather than mathematical intuition.

These manuscripts by Maxwell were not published at the time they were written despite Tait asking himto submit his ideas on knot theory to the Royal Society of Edinburgh for publication. However, more than100 years after they were written these manuscripts were published in [2]. There are three manuscripts onknots and some time between the second, which Maxwell wrote in October 1868, and the third, which hewrote on 29 December 1868, he had read Listing's 1847 paper Vorstudien zur Topologie for in the thirdmanuscript he lists Listing's main results. In February 1869 Maxwell presented an account of Listing'stopological ideas to the London Mathematical Society.

In 1869 Thomson tried to clarify the topological ideas that he was using. The problem really came downto the fact that, perhaps not surprisingly, he had confused what we know today are two different concepts.He wrote:-

I shall call a finite portion of space n-ply continuous when its bounding surface is such thatthere are n irreconcilable paths between any two points on it.

He explains that two irreconcilable paths between points P and Q are paths which cannot be smoothlytransformed into each other by paths which remain within the portion of space considered. This definitiondoes not really work although one can see Thomson struggling to reach a definition of homotopy. He doesnot compose his paths, however, a vital ingredient in the definition of homotopy. In the proofs he gave inthe paper Thomson does not even try to use his definition but rather resorts to arguments involving virtualbarriers which would stop fluid flowing. Again his topology is driven by physical ideas of fluid flow butthis notion, similar to the idea of a "cutting surface" which Riemann had introduced and Helmholtz hadused, relates to our present day idea of homology, not homotopy.

By 1876 Thomson had made little progress with his ideas of vortex atoms. There were many problems inhis way and indeed by this stage he had not succeeded in mathematically describing how two vortex ringswould interact if they did not have a common axis of symmetry much more the way that knotted vorticeswould interact. Also there was no insight into vortex atoms through lists of knots which, in Thomson'stheory, would explain the chemical elements. Tait decided to embark on a classification of plane closedcurves in 1876, writing in a report to the British Association for the Advancement of Science:-

The development of this subject promises absolutely endless work - but work of a veryinteresting and useful kind - because it is intimately connected with the theory of knots, which(especially as applied in Sir W Thomson's Theory of Vortex Atoms) is likely soon to becomean important branch of mathematics.

Now by looking at plane closed curves Tait was considering alternating knots, namely those which whentraversing the projection in 2-dimensional space the crossings go alternately over and under. Choosing astarting point and a direction to traverse the path, he labelled the first, third, fifth etc. points by A, B, Cetc. A knot with n crossings A, B, C, ... would then be described by the sequence of crossings of length 2nwhere each of A, B, C, ... occurred exactly twice when the knot was traversed. Tait called the sequencethe "scheme of the knot".

A knot with scheme ACBDCADB.


4 de 5 30/4/2007 16:18

There were then two basic problems to solve. Firstly which sequences of the above type correspond to aknot, and secondly how could it be determined when two knots described by such sequences were thesame. However there were some other problems, for example although a sequence of length 10, say,might represent a knot it might be one with less than 5 crossings. It might be a knot which could bereduced to one with fewer crossings. For example if the projection contained a crossing which divided thecurve into two parts which did not intersect, then this was a nugatory crossing which could be removedby a twist.

Example of a nugatory crossing.

Tait conjectured that an alternating diagram without nugatory crossings would contain the minimumnumber of crossings. This became known as Tait's first conjecture. He gave a "proof" which showed thatonly nugatory crossings allowed the number of crossings to be reduced. However this is not good enoughfor there might be a sequence of moves which first increase the number of crossings, then further movesreduce to a fewer number of crossings than were there originally. If we interpret Tait in a form that heseems to have used the conjecture, namely that two alternating diagrams without nugatory crossingsrepresenting the same prime knot are related by a sequence of twists, then we get what has been calledTait's second conjecture. This was not finally proved until 1993.

Without any rigorous theory, which would have been well beyond nineteenth century mathematics, Taitbegan to classify knots using his mathematical and geometrical intuition. He knew that what was reallyrequired was a knot invariant, that is something which would be independent of the way that the knot wasrepresented in two dimensions. First he looked for numerical invariants and considered the minimalnumber of crossings that a given knot might have in a two dimensional representation. This would leadhim to Tait's first conjecture for alternating knots.

Another idea which seemed promising to Tait was the "beknottedness" which he defined as follows.Travel round the knot diagram and immediately after each crossing throw a copper coin to the left and asilver coin to the right if the crossing was above, or throw a silver coin to the left and a copper coin to theright if the crossing was below.

This led to:

Tait then defined "beknottedness" (now known as the twist number) as the excess of silver crossings overcopper ones. If only diagrams without nugatory crossings were considered then Tait believed that this wasa knot invariant. In fact it is not, but for alternating knots, it is an invariant and this fact is a consequenceof Tait's second conjecture (a theorem since 1993). He tried other more obviously physical ideas such asconsidering the knot as a circuit and looking at the work done by a magnetic particle carried by a current


5 de 5 30/4/2007 16:18

in the knot. He tried another idea which at first looked very promising to him, namely the minimalnumber of crossings which required to be changed for under to over (or visa-versa) to unknot the knot.His first thought was that this would be half the beknottedness. He soon saw that this was not so. Seeingthat the two concepts were distinct Tait changed his definitions and called the minimal number ofcrossings which required to be changed to unknot the diagram the beknottedness and he called theminimal number of crossings the knottiness.

Listing had introduced polynomials in to variables associated with a knot. These were produced bymarking the four corners of a crossing l or r (for left of right) according to a rule which again related toover of under crossings. These polynomials were not invariants, however. Tait tried a similar idea wherehe marked the regions of the graph l or r and then connected r regions with a multiple bond of the order ofthe number of crossings on the boundary between the two regions. He was inspired to do this by analogywith chemical representations invented by his colleague (and brother-in-law) Alexander Crum Brown.

By 1877 Tait had classified all knots with seven crossings but he stopped there. He returned to the topicof knots in his address to the Edinburgh Mathematical Society in 1883:-

We find that it becomes a mere question of skilled labour to draw all the possible knotshaving any assigned number of crossings. The requisite labour increases with extremerapidity as the number of crossings is increased. ... I have not been able to find time to carryout this process further than the knots with seven crossings. ... It is greatly desired thatsomeone, with the requisite leisure, should try to extend this list, if possible up to 11 ...

Kirkman read the text of Tait's address and began to work on classifying knots with more than sevencrossings. He sent Tait his results on knot projections with up to nine crossings in May 1884 but he hadnot looked at the problem of deciding which of the projections led to equivalent knots. Tait worked onthis side of the problem and, considering only alternating knots, solved the equivalence problems within afew weeks. Tait seemed to know how to tell whether two knots were equivalent without rigorousmethods. He states this quite clearly in the paper he wrote tabulating the knots where he says that hismethods have:-

... the disadvantage of being to a greater or less extent tentative. Not that the rules laid down

... leave any room for mere guessing, but they are too complex to be always completely keptin view. Thus we cannot be absolutely certain that by means of such processes we haveobtained all the essentially different forms which the definition we employ comprehends.

Despite the problems Tait knew exactly what he was doing for, remarkably, his tables are correct. WhenKirkman sent him all knot projections with 10 crossings in January 1885 again Tait found all inequivalent knots. The tables were printed in September 1885 and again they are completely correct. Bythen he had received from Kirkman 1581 knot projections with 11 crossings and this time Tait felt that hedid not have the time to solve the equivalence problem for these. However by this time an Americanmathematician and engineer Charles N Little had sent Tait knot tables which he had calculated and Littlebegan to extend the tables to knots other than alternating ones, and to knots with eleven crossings.

Article by: J J O'Connor and E F Robertson

January 2001

MacTutor History of Mathematics[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Knots_and_physics.html]

Knots and Physics

Documents

Transcript of Knots and Physics