A revolution in knot theory (11/11/2011)
|This knot has Gauss code O1U2O3U1O2U3. - Graphic by Sam Nelson.|
In the 19th century, Lord Kelvin made the inspired
guess that elements are knots in the "ether". Hydrogen would be one
kind of knot, oxygen a different kind of knot---and so forth
throughout the periodic table of elements. This idea led Peter
Guthrie Tait to prepare meticulous and quite beautiful tables of
knots, in an effort to elucidate when two knots are truly different.
From the point of view of physics, Kelvin and Tait were on the wrong
track: the atomic viewpoint soon made the theory of ether obsolete.
But from the mathematical viewpoint, a gold mine had been discovered:
The branch of mathematics now known as "knot theory" has been
burgeoning ever since.
In his article "The Combinatorial Revolution in Knot Theory", to appear in the December 2011 issue of the Notices of the AMS,, Sam
Nelson describes a novel approach to knot theory that has gained
currency in the past several years and the mysterious new knot-like
objects discovered in the process.
As sailors have long known, many different kinds of knots are
possible; in fact, the variety is infinite. A *mathematical* knot can
be imagined as a knotted circle: Think of a pretzel, which is a
knotted circle of dough, or a rubber band, which is the "un-knot"
because it is not knotted. Mathematicians study the patterns,
symmetries, and asymmetries in knots and develop methods for
distinguishing when two knots are truly different.
Mathematically, one thinks of the string out of which a knot is formed
as being a one-dimensional object, and the knot itself lives in
three-dimensional space. Drawings of knots, like the ones done by
Tait, are projections of the knot onto a two-dimensional plane. In
such drawings, it is customary to draw over-and-under crossings of the
string as broken and unbroken lines. If three or more strands of the
knot are on top of each other at single point, we can move the strands
slightly without changing the knot so that every point on the plane
sits below at most two strands of the knot. A planar knot diagram is
a picture of a knot, drawn in a two-dimensional plane, in which every
point of the diagram represents at most two points in the knot.
Planar knot diagrams have long been used in mathematics as a way to
represent and study knots.
As Nelson reports in his article, mathematicians have devised various
ways to represent the information contained in knot diagrams. One
example is the Gauss code, which is a sequence of letters and numbers
wherein each crossing in the knot is assigned a number and the letter
O or U, depending on whether the crossing goes over or under. The
Gauss code for a simple knot might look like this: O1U2O3U1O2U3.
In the mid-1990s, mathematicians discovered something strange. There
are Gauss codes for which it is impossible to draw planar knot
diagrams but which nevertheless behave like knots in certain ways. In
particular, those codes, which Nelson calls *nonplanar Gauss codes*,
work perfectly well in certain formulas that are used to investigate
properties of knots. Nelson writes: "A planar Gauss code always
describes a [knot] in three-space; what kind of thing could a
nonplanar Gauss code be describing?" As it turns out, there are
"virtual knots" that have legitimate Gauss codes but do not correspond
to knots in three-dimensional space. These virtual knots can be
investigated by applying combinatorial techniques to knot diagrams.
Just as new horizons opened when people dared to consider what would
happen if -1 had a square root---and thereby discovered complex
numbers, which have since been thoroughly explored by mathematicians
and have become ubiquitous in physics and engineering---mathematicians
are finding that the equations they used to investigate regular knots
give rise to a whole universe of "generalized knots" that have their
own peculiar qualities. Although they seem esoteric at first, these
generalized knots turn out to have interpretations as familiar objects
in mathematics. "Moreover," Nelson writes, "classical knot theory
emerges as a special case of the new generalized knot theory."
Note: This story has been adapted from a news release issued by the American Mathematical Society