- High-Pitched Sounds Cause Seizures in Old Cats
- Body Clock's Molecular Reset Button
- Ice Bridge Migration Theory Wrong
- Strange Supernova Is 'Missing Link'
- Most Comprehensive Map of Universe
- Hate to Diet? It's How We're Wired
- Bizarre 'Platypus' Dinosaur Discovered
- Hot Vents Spontaneously Make Life's Molecules
- Is the Universe a Hologram?
- Human Flight to Mars With Electric Solar Sail

Science News

from research organizations

- Date:
- December 5, 2005
- Source:
- Rice University
- Summary:
- New research published in the mathematical journal Geometry and Topology describes a useful pattern of numbers that has lain hidden for almost 100 years in one of the most notable classification schemes for twisted knots. The pattern was discovered by Rice University mathematician Shelly Harvey. It was found in a widely used knot classification scheme that was originally described by the famed French mathematician Henri Poincaré.
- Share:

FULL STORY

The latest insight from Rice University assistant professor Shelly Harvey is the kind of idea that comes along rarely for a theorist in any discipline: It's an idea that is both simple and capable of explaining much.

The elegance of the idea and the breadth of its descriptive power are most readily apparent to mathematicians within Harvey's chosen discipline of topology. Harvey discovered an underlying structure – which went unnoticed for more than 100 years – within the mathematical descriptions that topologists most often use to characterize complex knots. The work was described in a paper that recently appeared in the journal Geometry and Topology.

"If someone comes up with a new mathematical theory that's 300 pages long with a lot of complex calculations, then you might suppose that the reason it hadn't been done previously was that it was too difficult," said fellow knot theorist and mathematics professor Tim Cochran. "However, real truth should be simpler and more beautiful than that, and this idea of Shelly's has the ring of truth to it. The moment I heard it, I knew she had hit upon something quite special."

Harvey's discovery applies to a longstanding problem within knot theory, but it can best be understood within the larger context of topology. Topology is a branch of math that's sometimes called "rubber sheet geometry" because topologists study objects that retain their spatial properties even when they are twisted into odd shapes. A classic example is the topological equivalence of a donut and a coffee cup. The donut can be stretched into the shape of the cup, where the hole in the center of the donut becomes the handle on the side of the cup. Thus the property of "having one hole" is preserved.

One of the underlying insights of topology is that some geometric problems depend not on the precise shape of objects but only on the way they are connected. In the classic example, 18th century mathematician Leonhard Euler proved that it was impossible to find a route through the Russian city of Königsberg that crossed each of the cities seven bridges just once. Topologically, the problem derives from the way the bridges connect the major islands of the city, so the result would be the same even if the primary shape of the town were – in the rubber-sheet analogy – twisted into a complex three-dimensional shape.

In knot theory, topologists are concerned with the spatial arrangements of unbroken lines that are folded in knots – not unlike a tangled kite string or fishing line. While the study of knots may sound esoteric, it does apply to real-world problems. DNA, for example, are long, unbroken strings of amino acids that fold naturally into complex, knotted clumps. The knotting and linking of strands of DNA is a biproduct of natural cellular processes and their unknotting is necessary for the cell to survive. It is known that enzymes dubbed topoisomerases have the job of unknotting those clumps, and topologists have been collaborating with cancer researchers in recent years to attempt to find novel cancer treatments that capitalize on that.

Topologists are keen to find ways to prove that two shapes, which may look very different, are truly inequivalent. One of the overarching goals in knot theory is to find a method that can determine equivalency in every case. Great attention has been paid to finding mathematical measures of a knot's complexity that can then be used to describe similarities and differences between knotted shapes. Sometimes these measures are actual numbers, like the so-called "unknotting number of a knot", and sometimes they are more sophisticated algebraic objects such as matrices or polynomials. One such measure developed 100 years ago by the Frenchman Henri Poincaré, which is reminiscent of Euler's Königsberg bridges problem, uses algebra to measure all possible paths that can be navigated in the space surrounding the knot, without ever touching the string itself. This collection of data is called the "fundamental group of the knot".

"I realized that there's an algebraic structure within the fundamental group of a knot. Some of these paths are more robust than others," Harvey said. "What Tim and I subsequently determined is that this structure remains unchanged as you try to unravel the knots. It even survives in four dimensions, which turns out to be a particularly handy tool for knot theorists because four-dimensional problems – like the jiggling of a DNA strand within a cell – happen to be some of the most difficult topological problems to understand."

Since Harvey's observation is so fundamental, it pertains as well to the study of many other topological objects, and these applications form part of her ongoing research program at Rice.

###

The research was funded by the National Science Foundation.

**Story Source:**

The above story is based on materials provided by **Rice University**. *Note: Materials may be edited for content and length.*

**Cite This Page**:

Rice University. "Mathematician's Insight Helps Unravel Knotty Problem." ScienceDaily. ScienceDaily, 5 December 2005. <www.sciencedaily.com/releases/2005/12/051205235631.htm>.

Rice University. (2005, December 5). Mathematician's Insight Helps Unravel Knotty Problem. *ScienceDaily*. Retrieved April 28, 2015 from www.sciencedaily.com/releases/2005/12/051205235631.htm

Rice University. "Mathematician's Insight Helps Unravel Knotty Problem." ScienceDaily. www.sciencedaily.com/releases/2005/12/051205235631.htm (accessed April 28, 2015).

Computers & Math News

April 28, 2015

Latest Headlines

updated 12:56 pm ET

Apr. 27, 2015 — Researchers have trained a computer to crunch big biomedical data in order to recognize how genes work together in human tissues. Combining genomic data from 38,000 experiments, this research group ... read more

Apr. 27, 2015 — Scientists have developed a revolutionary new technology that can image and weigh single molecules and instantly identify a single virus or bacteria ... read more

Apr. 27, 2015 — Researchers have captured the first 3-D video of a living algal embryo turning itself inside out, from a sphere to a mushroom shape and back again. The results could help unravel the mechanical ... read more

Apr. 27, 2015 — Technology can bolster efforts by parents, lawmakers and insurance companies to reduce distracted driving among novice teen drivers, according to a new ... read more

Apr. 25, 2015 — Studies have shown that patients who undergo surgeries on weekends tend to experience longer hospital stays and higher mortality rates and readmissions. For the first time, a new study has identified ... read more

Apr. 24, 2015 — Scientists have developed the first liquid nanoscale laser. And it's tunable in real time, meaning you can quickly and simply produce different colors, a unique and useful feature. The laser ... read more

Apr. 24, 2015 — Inspired by the Microsoft Kinect and the human eye, scientists have developed an inexpensive 3-D camera that can be used in any environment to produce high-quality ... read more

Apr. 24, 2015 — Using a novel microscopy technique, scientists revealed a major enhancement of coupling between electric and magnetic dipoles. The discovery could lead to devices for use in computer memory or ... read more