A nontrivial knot in 3-dimensional space
No. 144, 1974
(Knot theory)
India ink on paper, 31.5x44 cm.
A knot hovers in three-dimensional Euclidean space, dangling from a slim line like a spider preparing to quash its prey. But it is not just any knot. It is a nontrivial concrete knot, since it cannot be deformed by an isotopy into a standard circle embedded in a plane. Likewise, in topology, knot theory is truly nontrivial. That is so not only because of its inherent fascination and complexity, but also because of its incredible applications in biology and chemistry, where proteins and long polymer chains boil down to complex knots that can be understood and untangled. Given a certain knot, how can we figure out whether or not it is trivial? This is a terribly complicated problem, although an algorithm does exist to solve it. In some cases it is much easier to prove a knot's nontriviality. To do this, we look for the knots partial invariants to make computations.
