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THROUGH THE SALLYPORT

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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-

Foundation.

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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