Suppose someone hands you a loop of string with a big tangle in it. There is probably no mathematical way known for you to determine whether it can be untangled without cutting it.
Three members of the Department of Mathematics and Statistics at the University of Nevada, Reno, recently received a three-year grant from the National Science Foundation totaling more than $286,000, to look into this complicated problem.
According to one of the team members, Christopher Herald, an associate professor of mathematics, untying the mysteries behind what is known as "Knot Theory" has important implications for a number of different scientific fields. The most notable is in topology, part of the mathematical framework used by theoretical physicists to describe the universe.
"The thrust of our work is basic research, trying to understand knots and low-dimensional manifolds better in the abstract," Herald said, explaining that manifolds are theoretical objects, similar to surfaces, which are studied by mathematicians. "A lot of questions about manifolds can be boiled down to questions about knots, and that's the main motivation for the work we're doing.
"Sometimes biologists who are trying to understand how enzymes in cells manipulate DNA strands, or chemists constructing brand new substances by making knotted molecules, have questions about knots for the mathematicians. The tools that we develop will hopefully also help these scientists in the future."
While knots in three-dimensional space are concrete geometric objects—we can draw pictures of them—some of the most basic questions remain unsolved.
"For a particular knot, viewing it from different angles or stretching it into a slightly different shape will change the way it looks, and in particular will change how many times it appears to cross itself, so it's not clear what the simplest view of the knot is," Herald said. "Other difficult questions involve finding the simplest surface whose edge is the knot."
The study, being conducted by Herald, Swatee Naik, associate professor and Chair of the Department of Mathematics, and Stanislav Jabuka, assistant professor of mathematics, aims to study knot and surfaces using a powerful new set of techniques in low dimensional topology called the Heegaard Floer Homology.
"These techniques have been very successful at solving some difficult questions, but we are confident that, as we develop them further, they'll lead to more discoveries about knots and manifolds," Herald said.
"One of the things that is really enticing about this field is that many of the questions are really easy to explain, but are really hard to solve," Herald added, noting that a typical middle school math student can understand the question, "Can this tangled loop of string be rearranged, without cutting it, to look like its mirror image?"
"For simple examples, researchers have found clever ways to use abstract algebra and other mathematics to answer the question," Herald said. "But as soon as the knot is fairly complicated, for example, with more that 15 crossings, the question is wide open --hence the need for a study like the one the members of the math department are conducting."
"Even though we have this whole big tool box of ways to work on these problems, things can still be developed better," he added. "If we can improve mathematicians' knowledge of knots and surfaces, then it will be a successful study."