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A Knotty but
Useful Pattern
Knotty problems beg for explanations that are both elegant and broadly line. While the study of knots they are more sophisticated
descriptive—the kinds of ideas that aren’t easy to come by. But Rice
may sound esoteric, it does ap- algebraic objects like matrices
mathematician Shelly Harvey beat the odds when she discovered an
ply to real-world problems. or polynomials. One such mea-
underlying structure that had gone unnoticed for more than 100 years
DNA, for example, are long, sure developed 100 years ago
unbroken strings of nucleotides by Frenchman Henri Poincaré,
within the mathematical descriptions that topologists most often use to
that fold naturally into complex, which is reminiscent of Euler’s
characterize complex knots.
knotted clumps. The knotting Königsberg bridges problem,
and linking of strands of DNA is uses algebra to measure all pos-
a by-product of natural cellular sible paths that can be navigated
“If someone comes up with a “having one hole” is preserved. processes, and their unknotting in the space surrounding the
new mathematical theory that’s One of the underlying in- is necessary for the cell to sur- knot, without ever touching the
300 pages long with a lot of sights of topology is that some vive. It is known that enzymes string itself. This collection of
complex calculations, then you geometric problems depend not dubbed “topoisomerases” have data is called the “fundamental
might suppose that the reason on the precise shape of objects the job of unknotting those group of the knot.”
it hadn’t been done previously but only on the way they are clumps, and topologists have “I realized that there’s an
was that it was too diffi cult,” connected. In the classic ex- been collaborating with cancer algebraic structure within the
says Tim Cochran, professor of ample, 18th-century mathema- researchers in recent years to at- fundamental group of a knot,”
mathematics and fellow knot tician Leonhard Euler proved tempt to fi nd novel cancer treat- Harvey explains. “Some of these
theorist. “However, real truth that it was impossible to fi nd ments that capitalize on that. paths are more robust than
should be simpler and more a route through the Russian Topologists are keen to fi nd others. What Tim and I subse-
beautiful than that, and this idea quently determined is that this
of Shelly’s has the ring of truth structure remains unchanged
to it. The moment I heard it, I
One of the underlying insights of topology is that some
as you try to unravel the knots.
knew she had hit on something It even survives in four dimen-
quite special.”
geometric problems depend not on the precise shape of
sions, which turns out to be a
Harvey’s discovery applies objects but only on the way they are connected. particularly handy tool for knot
to a longstanding problem theorists because four-dimen-
within knot theory, but it can sional problems—like the jig-
best be understood within the
city of Königsberg that crossed ways to prove that two shapes
gling of a DNA strand within a
larger context of topology.
each of the city’s seven bridges that may look very different
cell—happen to be some of the
Topology is a branch of math
just once. Topologically, the are truly equivalent. One of
most diffi cult topological prob-
that is sometimes called “rub-
problem derives from the way the overarching goals in knot
lems to understand.”
ber-sheet geometry” because
the bridges connect the major theory is to fi nd a method that
Harvey’s observation is so
topologists study objects that
islands of the city, so the re- can determine equivalency in
fundamental that it pertains
retain their spatial properties
sult would be the same even if every case. Great attention has
to the study of many other to-
even when they are twisted into
the primary shape of the town been paid to fi nding math-
pological objects, and these
odd shapes. A classic example
were—in the rubber sheet anal- ematical measures of a knot’s
applications form part of her
is the topological equivalent of
ogy—twisted into a complex complexity that can then be
ongoing research at Rice. The
a donut and a coffee cup. The
three-dimensional shape. used to describe similarities and
research was described in the
donut could be stretched into
In knot theory, topologists differences between knotted
November issue of the journal
the shape of the cup, where the
are concerned with the spatial shapes. Sometimes these mea-
Geometry and Topology and was
hole in the center of the donut
arrangements of unbroken lines sures are actual numbers, like
funded by the National Science
becomes the handle on the side
that are folded in knots, like the so-called “unknotting num-
of the cup. Thus the property of
tangled kite string or fi shing ber of a knot,” and sometimes —Jade Boyd
Spring ’06 15
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