Peter Guthrie Tait

How to use this course

Peter Guthrie Tait is remembered in three separate communities that rarely talk to one another. Physicists know “Thomson and Tait”, the treatise that taught energy methods to a generation. Knot theorists know the Tait conjectures, settled a century after he made them. Graph theorists know Tait colourings and Tait’s Hamiltonian conjecture. This course connects them. The thread is Victorian mathematical physics in Scotland, where a speculative theory of atoms led to the first systematic mathematics of knots.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Where a modern result settled a question Tait raised, the lesson gives both his formulation and the modern proof.

Timeline

Dalkeith and the Academy, 1831–1847

Peter Guthrie Tait was born on 28 April 1831 at Dalkeith, a market town south-east of Edinburgh. His father, John Tait, was secretary to the fifth Duke of Buccleuch; his mother was Mary Ronaldson. He was the only son among three children. He started at the grammar school in Dalkeith, but his father died when Peter was about six, and his mother moved the family into Edinburgh to live with her brother, the banker John Ronaldson. The uncle shared his interests in geology, astronomy and photography with the boy. At about thirteen Tait followed the satellites of Jupiter for two weeks and charted their motions.

After a private school in Circus Place he entered the Edinburgh Academy in 1841, aged ten. Maxwell was a year ahead of him there, and the two began a friendship that lasted until Maxwell’s death in 1879. Tait was top of his class in each of his six years. In the Academy’s mathematics prize competitions, Tait placed first in 1846 with Maxwell third; in 1847 Maxwell was first and Tait second.

In November 1847, aged sixteen, he entered the University of Edinburgh with Maxwell. He took mathematics under Philip Kelland and natural philosophy under James David Forbes, the glaciologist and experimenter. Forbes put him to calculating wooden discs for models of the catenary. A year later he moved south to Cambridge.

Peterhouse and Belfast, 1848–1860

Tait went to Peterhouse, Cambridge, in 1848, where William Hopkins, the most famous mathematical coach of his time, prepared him for the Tripos. In January 1852 he was Senior Wrangler and first Smith’s Prizeman, at the age of twenty. Peterhouse elected him a Fellow, and he coached undergraduates. With William Steele he planned a textbook of particle dynamics; Steele died early, and Tait wrote most of A Treatise on Dynamics of a Particle (1856) himself.

Belfast

In September 1854 he became Professor of Mathematics at Queen’s College, Belfast. There he met the chemist Thomas Andrews, later known for the critical point of carbon dioxide, and did his first experimental work with him, on the density of ozone and the effect of electrical discharge on oxygen and other gases. He married Margaret Archer Porter in Belfast on 13 October 1857. Her brothers had been Tait’s friends; her sister later married the Edinburgh chemist Alexander Crum Brown.

Tait had read Hamilton’s Lectures on Quaternions in 1853. In 1858, after reading Hermann von Helmholtz’s paper on vortex motion, he saw that quaternions could express its equations compactly, and he wrote to Hamilton in Dublin. The correspondence that followed made him Hamilton’s chief disciple. Lesson 04 describes the mathematics.

The Edinburgh chair, 1860–1879

In 1859 Forbes left the Edinburgh Chair of Natural Philosophy to become Principal at St Andrews. Among the candidates were Tait and Maxwell, who was about to lose his Aberdeen post in a merger of colleges. The curators chose Tait. A local newspaper explained that the committee valued his skill in explaining physics to students with little preparation. Maxwell went to King’s College London instead, and the two friends kept up a lively correspondence, signing letters with jokes and abbreviations. Maxwell called Tait “T′” and Thomson “T”, and signed himself dp/dt, from an equation in Tait’s thermodynamics.

Tait took up the chair in 1860 and held it for forty-one years. He was a gifted lecturer, with large classes and spectacular demonstrations; the novelist J. M. Barrie, an Edinburgh student, left a vivid sketch of him at the lecture table. In 1868 he opened a physical laboratory for students, modelled on Thomson’s laboratory in Glasgow, and enlarged it in 1880 by taking over former dissecting rooms. Among his laboratory assistants were William Robertson Smith (1868–70), later a famous biblical scholar, and Alexander Macfarlane, who took his doctorate in 1878 with a method Tait suggested and later carried the quaternion cause to America.

He was secretary of the Royal Society of Edinburgh from 1864 and its General Secretary from 1879 until his death. Many of his roughly 365 papers and reviews appeared in its Proceedings and Transactions. He was never elected a Fellow of the Royal Society of London.

Quaternions and the nabla

For the general reader. Complex numbers describe rotations in the plane. In 1843 Hamilton found a four-dimensional number system, the quaternions, that does the same for rotations in space, at the price of giving up the rule \(ab=ba\). Tait spent his life showing physicists how to use them. Many of the operations of vector calculus, such as the divergence and curl, first appeared together in this quaternion form.

The algebra

A quaternion is \(q=w+xi+yj+zk\) with real \(w,x,y,z\) and Hamilton’s rules

Hamilton called \(w\) the scalar part \(Sq\) and \(xi+yj+zk\) the vector part \(Vq\). Identify a vector \(\mathbf a\in\mathbb R^3\) with the pure quaternion \(a_1i+a_2j+a_3k\). Expanding with the rules gives the key identity

In particular \(\alpha^2=-|\mathbf a|^2\): every unit vector is a square root of \(-1\). The conjugate \(\bar q=w-xi-yj-zk\) satisfies \(q\bar q=|q|^2\), so every non-zero quaternion has an inverse \(\bar q/|q|^2\). The quaternions form a division algebra, and by Frobenius’s theorem of 1877 they are the only finite-dimensional associative real division algebra besides \(\mathbb R\) and \(\mathbb C\).

Sketch. Write \(v=v_\parallel+v_\perp\) relative to \(\mathbf n\). Since \(\mathbf n\) commutes with \(v_\parallel\), \(qv_\parallel q^{-1}=v_\parallel\). Since \(\mathbf n\) anticommutes with \(v_\perp\), \(qv_\perp\bar q=q^2v_\perp=(\cos\theta+\mathbf n\sin\theta)v_\perp=\cos\theta\,v_\perp+\sin\theta\,\mathbf n\times v_\perp\), the rotated vector. The half angles explain the double cover: \(q\) and \(-q\) give the same rotation.

Hamilton’s operator

Hamilton introduced the operator \(\nabla=i\,\partial_x+j\,\partial_y+k\,\partial_z\). Tait made it a working tool for physics in his treatises, and the name nabla, after a Hebrew harp of similar shape, was suggested to him by Robertson Smith. Applied to a vector field \(\sigma=ui+vj+wk\), the quaternion product gives

Maxwell, reading Tait, coined the words convergence (for \(-\operatorname{div}\)) and curl in 1870, and used quaternion notation alongside Cartesian components in his Treatise on Electricity and Magnetism of 1873. In quaternion form two of Maxwell’s equations combine into a single equation for \(\nabla\).

T and T′: the Treatise on Natural Philosophy

For the general reader. In 1861 Tait began a textbook of mathematical physics, and William Thomson, the future Lord Kelvin, joined him. Their Treatise on Natural Philosophy (1867) built all of mechanics on energy and on the methods of Lagrange and Hamilton. Physicists called it “T and T′” after the authors’ initials. It shaped how British physicists thought for half a century.

The book

The first edition (1867) was one volume of a planned four; an enlarged second edition appeared in two parts (1879, 1883), and the later volumes were never written. Tait wrote most of the text. The book made Lagrange’s generalized coordinates, Hamilton’s principle and the conservation of energy the frame for every topic from the elastic rod to the tides. A shorter Elements of Natural Philosophy (1872) served students.

Gyroscopic stability

Linearize a mechanical system with velocity-dependent but non-dissipative forces, such as Coriolis forces in a rotating frame or the forces in a spinning top, about an equilibrium:

The antisymmetric term \(G\dot q\) does no work: \(\dot q^{\mathsf T}G\dot q=0\). Without it, the equilibrium is stable exactly when \(K\) is positive definite. The Treatise asked when gyroscopic forces can stabilize an equilibrium that is unstable without them.

Proof of the first part. Try \(q=u\,e^{\lambda t}\). Then \(\lambda\) must be a root of \(p(\lambda)=\det(\lambda^2M+\lambda G+K)\). Transposing the matrix replaces \(G\) by \(-G\), which is the same as replacing \(\lambda\) by \(-\lambda\); so \(p(-\lambda)=p(\lambda)\), and \(p\) is a polynomial in \(\lambda^2\) of degree \(2n\) with leading coefficient \(\det M>0\). At \(\lambda=0\), \(p(0)=\det K\), which is negative because \(K\) has an odd number of negative eigenvalues. By the intermediate value theorem \(p\) has a real root \(\lambda>0\), giving an exponentially growing solution. The second part, proved rigorously later by Nikolai Chetaev, is why a spinning top eventually falls.

Smoke rings and vortex atoms

For the general reader. A smoke ring holds its shape as it crosses a room and bounces off another ring as if it were solid. In 1858 Helmholtz proved that in a perfect fluid a vortex ring can never be created or destroyed. Tait built a box that fired smoke rings, and Thomson, watching them in 1867, proposed that atoms might be knotted vortex rings in the ether. The theory failed, but it made knots a subject of physics.

Helmholtz and Kelvin

For a velocity field \(\mathbf u\), the vorticity is \(\boldsymbol\omega=\nabla\times\mathbf u\) and the circulation around a closed curve \(C\) is \(\Gamma=\oint_C\mathbf u\cdot d\mathbf l=\int_S\boldsymbol\omega\cdot d\mathbf S\).

Sketch of Kelvin’s theorem. For a material curve, \(\frac{d}{dt}\oint\mathbf u\cdot d\mathbf l=\oint\frac{D\mathbf u}{Dt}\cdot d\mathbf l+\oint\mathbf u\cdot d\mathbf u\). The second integral is \(\oint d(\tfrac12|\mathbf u|^2)=0\). By Euler’s equation, \(D\mathbf u/Dt=-\nabla p/\rho-\nabla\Phi\); for barotropic flow \(\nabla p/\rho=\nabla h\) for an enthalpy \(h(p)\), so the first integral is the integral of a gradient around a closed loop, also zero.

Tait’s smoke box and the vortex atom

Early in 1867 Tait built a box with a circular hole in one face, filled it with smoke from ammonia and hydrochloric acid, and struck the opposite face to launch rings. He showed Thomson how rings rebound and vibrate. Thomson’s paper On vortex atoms (1867) followed at once: if the ether is a perfect fluid, Helmholtz’s theorems make a knotted vortex permanent, and its knot type cannot change. Different chemical elements might be different knots, and their spectral lines the vibrations of the rings. The programme needed a list of knots. That was the job Tait took on (Lesson 07).

A thin ring of radius \(R\), core radius \(a\ll R\) and circulation \(\Gamma\), with uniform vorticity in the core, moves along its axis at Kelvin’s speed

Knot tables and the Tait conjectures

For the general reader. When are two tangled loops of string the same knot? Tait drew every knot he could with up to seven crossings, and later, with help from a clergyman mathematician, up to ten, sorting out which drawings showed the same knot. Along the way he guessed several rules about the simplest drawings. Those guesses resisted proof for a hundred years, until a new invariant discovered in 1984 settled them.

The tables

Tait published seven papers on knots in the Proceedings of the Royal Society of Edinburgh in 1876–77 and by 1877 had classified knots with up to seven crossings. In an 1883 address he asked for the tables to be extended. The Rev. Thomas Kirkman sent him all the relevant diagrams with eight and nine crossings in 1884 and ten in 1885, and Tait sorted the alternating ones into knot types. Charles Little, in the 1890s, completed the non-alternating ten-crossing knots. The tables were remarkably accurate: one duplicated pair in Little’s list, the Perko pair, was found only in 1974. Tait also coined much of the language, including knottiness for what is now called the crossing number.

Definitions

A knot diagram is a projection of a knot to the plane with finitely many transverse double points, each marked over or under. A diagram is alternating if the crossings alternate over and under along every strand, and reduced if no crossing can be removed by twisting part of the diagram (no nugatory crossing). The crossing number \(c(K)\) is the fewest crossings in any diagram of \(K\). For an oriented diagram, each crossing has a sign \(\pm1\), and the writhe \(w(D)\) is their sum.

The bracket proof of the first conjecture

Kauffman’s bracket assigns to an unoriented diagram \(D\) a Laurent polynomial \(\langle D\rangle\in\mathbb Z[A,A^{-1}]\). Each crossing can be smoothed in two ways, called the A- and B-smoothings. A state \(s\) chooses one smoothing at every crossing and produces \(|s|\) disjoint circles. Then

where \(a(s),b(s)\) count the A- and B-smoothings. It is invariant under Reidemeister moves II and III and changes by \(-A^{\pm3}\) under move I; the normalized \(V(t)=(-A^3)^{-w(D)}\langle D\rangle\) with \(A=t^{-1/4}\) is the Jones polynomial.

Let \(s_A\) and \(s_B\) be the all-A and all-B states of a connected diagram with \(n\) crossings. One checks three facts. The highest power of \(A\) in \(\langle D\rangle\) is at most \(n+2|s_A|-2\) and the lowest at least \(-n-2|s_B|+2\), so the span satisfies \(\operatorname{span}\langle D\rangle\le2n+2(|s_A|+|s_B|)-4\). For any connected diagram \(|s_A|+|s_B|\le n+2\), with equality for alternating diagrams (the circles of \(s_A\) and \(s_B\) are the boundaries of the black and white regions of the checkerboard colouring, and Euler’s formula counts \(n+2\) regions). Finally, for a reduced alternating diagram the extreme coefficients are \(\pm1\) and do not cancel. So

Since \(\operatorname{span}V_K\) depends only on \(K\), no diagram can have fewer crossings than a reduced alternating one.

Four colours and three edge colours

For the general reader. Can every map be coloured with four colours so that neighbouring countries differ? In 1879 Alfred Kempe published a proof, and in 1880 Tait published several more. All were wrong. Tait’s lasting contribution was a translation: colouring the countries of a map with four colours is the same as colouring the borders with three. The translation remains one of the standard ways to think about the problem.

It suffices to colour maps in which exactly three countries meet at each point: the boundary graph is then a bridgeless planar cubic graph (every vertex of degree 3). A Tait colouring is a colouring of the edges with three colours so that the three edges at each vertex all differ.

Proof. Use the Klein four-group \(\{0,a,b,c\}\) with \(a+b=c\) and \(x+x=0\). Given a face colouring with these four elements, colour each edge by the sum of the colours of the two faces it separates; this is non-zero because neighbouring faces differ. At a vertex the three faces have distinct colours \(x,y,z\), and the edges get \(x+y,\ y+z,\ x+z\), which are distinct. Conversely, given a Tait colouring by \(a,b,c\), colour a fixed face 0 and give any other face the sum of the edge colours crossed by a path to it. The result is independent of the path, because around each vertex the three colours sum to \(a+b+c=0\), and in the plane every closed path is a sum of loops around vertices. Adjacent faces differ by a non-zero edge colour. ∎

Tait thought every bridgeless cubic planar graph was easy to edge-colour and gave faulty arguments. Julius Petersen showed in 1891 that one of them was incorrect. The theorem was finally proved, by computer, by Kenneth Appel and Wolfgang Haken in 1976. Non-planar bridgeless cubic graphs without Tait colourings, now called snarks, exist; the smallest is Petersen’s graph. In this language the four colour theorem says that no snark is planar.

Tait’s Hamiltonian conjecture (1884)

A Hamiltonian cycle passes through every vertex exactly once. If a cubic graph has one, it has an even number of vertices, so colouring the cycle’s edges alternately with two colours and the remaining edges with the third gives a Tait colouring. Tait conjectured that every 3-connected cubic planar graph has a Hamiltonian cycle, which would have proved the four colour theorem. William Tutte found a counterexample with 46 vertices in 1946.

Thermoelectricity and energy

For the general reader. Join two different metals in a loop and heat one junction: a current flows. Tait measured this effect for more than twenty metals over a wide range of temperatures and found a surprisingly simple pattern, which he displayed in a diagram that became standard in textbooks. He was also, with Thomson, one of the loudest promoters of the new science of energy.

The Thomson relations

For a thermocouple of metals A and B with junctions at \(T_0\) and \(T\), let \(E(T)\) be the electromotive force and \(P_{AB}=dE/dT\) the thermoelectric power (Seebeck coefficient). In the 1850s Thomson, applying thermodynamics to the reversible heat effects, derived

where \(\Pi_{AB}\) is the Peltier coefficient (heat absorbed per unit charge at a junction) and \(\sigma\) the Thomson coefficient of each metal (heat absorbed per unit charge per degree along a conductor with a temperature gradient).

Tait plotted \(P\) against \(T\) for each metal. On this thermoelectric diagram each metal is a straight line, and the e.m.f. of a couple between \(T_0\) and \(T\) is the area between the two lines over that interval: \(E=\int_{T_0}^{T}(P_A-P_B)\,dT\). Where two lines cross, at the neutral temperature \(T_n\), the couple’s e.m.f. is maximal. He presented the diagram in his 1873 Rede Lecture at Cambridge.

Energy

Tait wrote one of the early textbooks of thermodynamics and, with Thomson, publicized the principle of energy and its dissipation in lectures, reviews and articles. In the 1880s he turned to the kinetic theory of gases. A series of papers from 1886 to 1892 included a proof of the equipartition of energy among colliding spheres that Thomson regarded as the first rigorous one.

Deep-sea pressure and the flight of a golf ball

For the general reader. Tait enjoyed hard practical problems. The Challenger expedition (1872–76) had measured temperatures miles down in the ocean, but the crushing pressure distorted the thermometers. Tait measured how water and glass compress and found a simple formula still used, in modified form, today. In his last decade, with a champion golfer for a son, he explained why a well-struck golf ball flies so far: backspin makes it lift.

The Tait equation

Between 1879 and 1888 Tait built apparatus to subject thermometers and liquids to pressures of several hundred atmospheres. His report for the Challenger volumes (1881 and later) gave the pressure correction for the deep-sea thermometers and an empirical law for the compression of water. In his form, the average compressibility over a pressure \(p\) is

Spin and lift

Golfers had found that a ball with a worn, rough surface flew further than a new smooth one, and that a well-struck drive seemed to rise rather than fall along a parabola. In articles in Nature from 1890 and a paper on the path of a rotating spherical projectile (1896), Tait argued that underspin produces an upward force by the effect Heinrich Gustav Magnus had described for spinning cylinders in 1853. Rough surfaces grip the air better and increase the effect. His son Freddie, Amateur Champion in 1896 and 1898, supplied the practical tests.

In modern notation, a ball of radius \(r\) moving at speed \(v\) through air of density \(\rho\) feels drag and lift

where the lift coefficient \(C_L\) grows with the spin parameter \(r\omega/v\), and lift acts perpendicular to the velocity, upward for backspin.

Controversies and the vector war

Tait was combative in print. In the 1860s he fought a long priority dispute with John Tyndall over the mechanical equivalent of heat: Tyndall championed Julius Robert Mayer, while Tait and Thomson championed James Prescott Joule. Later Tait quarrelled with Rudolf Clausius over the history of thermodynamics. Historians judge his accounts of that history partisan, slanted toward British claims.

Quaternions against vectors

After Hamilton’s death in 1865, Tait led the effort to make quaternions the language of physics. Others extracted what they needed and discarded the rest. Josiah Willard Gibbs, in privately printed notes of 1881–84, and Oliver Heaviside, in his electromagnetic papers, separated the product \(\alpha\beta\) into a dot product and a cross product and dropped the four-dimensional algebra. In 1890–93 the dispute ran in the pages of Nature. Tait called Gibbs’s system a hybrid that lost the essential associativity and division of quaternions; Gibbs and Heaviside replied that physicists needed scalars and vectors, and that the full quaternion product rarely had a physical meaning.

The vectors won. By 1910 most physics was written in Gibbs–Heaviside notation, and quaternions survived mainly through Tait’s student Alexander Macfarlane and the international Quaternion Association. The story did not end there. Unit quaternions are the spin group \(\mathrm{SU}(2)\), the double cover of rotations (Lesson 04); they describe the spin of the electron, and they are now the standard way to store rotations in spacecraft attitude control and computer graphics.

The Unseen Universe

In 1875 Tait and Balfour Stewart published, at first anonymously, The Unseen Universe, arguing that the conservation and dissipation of energy were compatible with, and even suggested, an invisible world beyond the physical one. The book was aimed against scientific materialism and sold widely. A sequel, Paradoxical Philosophy, followed in 1878.

Last years and legacy, 1890–1901

Tait received the Royal Medal of the Royal Society of London in 1886, the Keith Prize of the Royal Society of Edinburgh twice and its Gunning Victoria Jubilee Prize, and honorary degrees from Glasgow and the Catholic University of Ireland. His Scientific Papers appeared in two volumes in 1898 and 1900.

He and Margaret had seven children. Their third son, Frederick Guthrie Tait, known as Freddie, was the leading amateur golfer in Britain, winning the Amateur Championship in 1896 and 1898. He served with the Black Watch in South Africa, was wounded at Magersfontein in December 1899, and was killed leading an attack at Koodoosberg on 7 February 1900. His father never recovered from the blow. Tait retired from his chair early in 1901 and died a few months later, on 4 July 1901, at Challenger Lodge in Wardie, near Leith. He was buried at St John’s Episcopal Church, Edinburgh. His former assistant Cargill Gilston Knott published Life and Scientific Work of Peter Guthrie Tait in 1911.

Legacy

• Knot theory: his tables, terminology and conjectures founded the subject. Their resolution in 1987–1993 through the Jones polynomial was one of the triumphs of modern low-dimensional topology, and vortex knots have returned to physics in fluids, plasmas and superfluids.
• Graph theory: Tait colourings, the parity lemma and Tait’s Hamiltonian conjecture remain basic tools and examples.
• Physics teaching: the Treatise on Natural Philosophy made energy and Lagrangian methods the core of British physics; his laboratory was a model for practical teaching.
• Quaternions: lost the notation war but won in the long run, as the algebra of spin and of rotations in engineering.
• Applied physics: the Tait equation and the physics of spinning balls are still in use.

The University of Edinburgh named a chair in physics after him, and Peter Guthrie Tait Road runs through its King’s Buildings campus.

Connections

People linked to Peter Guthrie Tait across this collection. A name without a link has no page here yet. See Peter Guthrie Tait in the network of all 337 lives →

Teachers and advisors

• James David Forbes
• Philip Kelland
• William Hopkins 1848-1852

Students

• Alexander Macfarlane 1874-1878
• William Robertson Smith 1868-1870

Influenced by

• Hermann von Helmholtz
• William Rowan Hamilton 1858-1865

Influenced

• Louis Kauffman
• Morwen Thistlethwaite
• William Tutte

Colleagues and collaborators

• George Gabriel Stokes
• James Prescott Joule
• William Thomson, Lord Kelvin 1861-1901
• Balfour Stewart 1866-1878
• Cargill Gilston Knott
• James Dewar
• Thomas Andrews 1854-1860
• Thomas Kirkman 1884-1885
• William John Steele

Rivals

• John Tyndall
• Josiah Willard Gibbs 1890-1893
• Julius Robert Mayer 1862
• Oliver Heaviside
• Rudolf Clausius

Friends

• James Clerk Maxwell 1841-1879

Family

• Alexander Crum Brown

Also mentioned in this life

• Gaspard-Gustave Coriolis
• Leonhard Euler
• Georg Frobenius
• Felix Klein
• Joseph-Louis Lagrange

Fields

• mathematical physics 45 lives
• knot theory 3 lives
• graph theory 4 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Peter Guthrie Tait, MacTutor History of Mathematics.
2. Tait, Peter Guthrie, Dictionary of National Biography entry, via MacTutor.
3. Peter Guthrie Tait, Wikipedia.
4. Peter Guthrie Tait and the Scrapbook, MacTutor.
5. C. G. Knott, Life and Scientific Work of Peter Guthrie Tait, Cambridge University Press, 1911.
6. Peter Guthrie Tait, booklet, James Clerk Maxwell Foundation.

Mathematics and physics

1. W. Thomson and P. G. Tait, Treatise on Natural Philosophy, Oxford, 1867; 2nd ed. Cambridge, 1879–83.
2. P. G. Tait, An Elementary Treatise on Quaternions, Oxford, 1867.
3. P. G. Tait, On knots, I–III, Trans. R. Soc. Edinburgh 28 (1877), 32 (1884–85); reprinted in Scientific Papers, vol. 1, Cambridge, 1898.
4. P. G. Tait, Remarks on the colouring of maps, Proc. R. Soc. Edinburgh 10 (1880).
5. P. G. Tait, Report on some of the physical properties of fresh water and of sea water, Report on the Scientific Results of the Voyage of H.M.S. Challenger, Physics and Chemistry, vol. 2 (1888).
6. W. Thomson, On vortex atoms, Proc. R. Soc. Edinburgh 6 (1867).
7. Tait conjectures, Wikipedia (with references to Kauffman, Murasugi, Thistlethwaite and Menasco).
8. L. H. Kauffman, State models and the Jones polynomial, Topology 26 (1987).
9. W. Menasco and M. Thistlethwaite, The classification of alternating links, Annals of Mathematics 138 (1993).
10. W. T. Tutte, On Hamiltonian circuits, J. London Math. Soc. 21 (1946).
11. M. J. Crowe, A History of Vector Analysis, Notre Dame, 1967; Dover reprint.
12. D. S. Silver, Knot theory’s odd origins, American Scientist 94 (2006).
13. C. Adams, The Knot Book, W. H. Freeman, 1994.