A Circular History of Knot Theory
In the nineteenth century physicists were speculating about the underlying
principles of atoms. In 1867, Lord Kelvin put forward a comprehensive theory
of atoms which, through heuristic reasoning, seemed to explain several of
the essential qualities of the chemical elements. Kelvin's theory conjectured
that atoms were knotted tubes of ether. (To a topologist a knot in 3-space is any closed loop having no self-intersections and a link is any
collection of non-intersecting closed loops.) The topological stability
and the variety of knots were thought to mirror the stability of matter
and the variety of chemical elements.
Kelvin's theory of vortex atoms was taken seriously for about two decades.
Maxwell thought that ``it satisfies more of the conditions than any atom
hitherto considers''. This theory inspired the celebrated Scottish physicist
Peter Tait to undertake an extensive study and tabulation of knots in an
attempt to understand when two knots were ``different''. (The later stages
of this study were in collaboration with C. N. Little.) Tait's intuitive
understanding of ``different'' and ``same'' is still a useful notion. Two
knots are isotopic if one can be continuously manipulated in 3-space (no self-intersections allowed) until it looks like the other. The
accompanying diagram shows a portion of Tait's study---an enumeration of knots and links in
terms of the crossing number of a plane projection. If Kelvin's theory had
been the correct foundation for the classification of the chemical elements,
then Tait's knot table would have been the basis for a periodic table of
elements. But Kelvin's theory was fundamentally mistaken and physicists
lost interest in the Tait's work.
What the physicists abandoned, intrigued mathematicians, then and now, and
the basic question is still the same: how do we tell when two knots are isotopically the same? (Research tip: Sometimes the most interesting problems can be found in
someone else's trash.) This failed atomic theory also left in its wake the
riches of Tait's tabulation---163 knot projections---and a rudimentary understanding
of isotopic sameness in terms of how one projection could be continuously
manipulated to look like another. This understanding of projection manipulation
was summarized in a set of conjectures for knot projections, the famous Tait Conjectures.
To attack the Tait Conjectures and the basic question of sameness of knots,
topologists developed knot invariants. An early example of a successful knot invariant is the Alexander polynomial, discovered by J. W. Alexander in 1927. The Alexander polynomial for the
knot labeled 3_1 (the trefoil) is -(txt)+t -1 and the polynomial for 4_1 (the figure-eight) is -(txt)+3t-1. Since these two polynomials are different we know their associated knots
are different. The Alexander polynomial was remarkable for how successful
it was in distinguishing the knots in Tait's orginal table and it gave witness
to how thorough a researcher Tait was. (Historical note: The last of the
few duplications in the Tait/Little table was found in 1974 by Kenneth Perko,
a New York lawyer and part-time topologist, while he was manipulating loops
of rope on his living room floor. If a lawyer can do research in knot theory,
it can't be that hard.) Unforunately, there are many knots with equivalent
Alexander polynomial that can be shown to be isotopically different through
the uses of other invariants.
So the search was on for more sensitive knot invariants that would detect
when two knots were different. This led to alternate understandings of the
notion of sameness. In particular, to a topologist there is no difference
between the loops representing 4_1 and 5_1. What is different is the space away from these loops, that is the complement of the knot. Two topological spaces are homeomorphic if there is a bijective invertible continuous function that maps one space
to the other. Thus, we have an alternate notion of sameness: if two knots/links
have homeomorphic knot/link complements then they are homeomorphic knots/links. Now, it would seem that homeomorphic sameness would be weaker than isotopic
sameness. And in fact, for link complements it is---there exist examples
of links that are not isotopic, but have homeomorphic complements. But for
knots a seminal result of Cameron Gordon and John Luecke showed that two
knot are homeomorphic if and only if they are isotopic. In the vernacular
of the knot theorist, a knot determines its complement.
Understanding that the principle object of study is the knot complement
places knot theory inside the larger study of 3-manifolds. A 3-manifold is a space which locally (assume you are near sighted) looks like
standard xyz-space and knot complements are readily seen as examples of 3-manifolds. It was through the study of 3-manifolds that in the 1970's knot theory began returning to its ancestoral
roots in physics. To understand this we have to flashback to the 1860's
work of Bernhard Riemann. Riemann was interested in relating geometric structures
to the forces in physics. Building on Gauss' work, Riemann investigated
three different geometric structures for 3-dimensional spaces---elliptic, euclidean, and hyperbolic. (Einstein's Theory
of Relativity was built on Riemannian geometry.) Each of these distinct
structures can be characterized by the behavior of triangles in planes.
In elliptic 3-space, the interior angles of a triangle in a plane have a sum greater than 180 degrees. In Euclidean 3-space, the sum is 180 degrees and in hyperbolic 3-space the sum is less than 180 degrees. In 1978, William Thurston established sufficient conditions for
when a 3-manifold possesses a hyperbolic structure. Surprisingly, except for a well
understood subclass of knots, all knot complements possess a complete hyperbolic
structure. (The beauty of Thurston's work is captured in the video Not Knot that is distributed by the American Mathematical Society and has been frequently
viewed at Grateful Dead concerts.)
Thurston's work on hyperbolic structures firmly re-established knot theory's
connections with physics. In the 1980's, through some totally unexpected
routes, knot theory made further connections with its ancestral roots. In
1987 Vaughan Jones discovered a totally different polynomial invariant from
that of Alexander using the theory of operator algebras. Within a short
period of time, more than five new polynomial invariants generalizing the
Jones polynomial were discovered. (One of these polynomial was simultaneously
discovered by six different mathematicians and its name is an acronym of
their last names---HOMFLY.) Moreover, Jones' polynomial quickly led to the
proofs that established all of Tait's original conjectures on knot projections.
With this poliferation of new polynomials it was natural to ask whether
any of these invariants had a natural extensions to all 3-manifolds. Two facts worked in favor of having such extensions: 1) all 3-manifolds can be describe in terms of knots and links via an operation called Dehn surgery; 2) there exists a set of moves, the Kirby calculus, that allow one to
move between differing Dehn surgery descriptions of the same homeomorphic 3-manifold. Using the Kirby calculus as a means to generalizing the polynomial
invariants, Edward Witten, a theoretical physicist, proposed new invariants
for 3-manifolds. His invariants came out of the theoretical area of physics know
as quantum field theory. These new invariants can be realized as certain
averages of link polynomials obtained from a given Dehn surgery representation
of the manifold.
Starting with the flawed theory of Kelvin's knotted vortex to the work of
Thurston, Jones and Witten, knot theory has circled back to its ancestral
orgins of theoretical physics.
Note; If you are interested in reading more about Knot Theory and 3-manifolds, Dale Rolfsen's book, Knots and Links, is a good introductory source.