How is a doughnut like a coffee cup, but different from a coiled spring? The riddle is key to understanding this year’s Nobel Prize in Physics, which honors theorists David Thouless of the University of Washington, Seattle, Michael Kosterlitz of Brown University, and Duncan Haldane of Princeton University.

The answer is that because each has one hole, the doughnut and the coffee cup have the same topology. An infinitely pliable cup could be molded into a doughnut without tearing it. The spring, on the other hand, can be unwound into a wire, flattened into a sheet, or squished into a ball. But it cannot be smoothly deformed to a make a doughnut. Thus, it has a different “topological charge”—essentially, the number of holes. Thouless, Kosterlitz, and Haldane showed that topology is more than a mathematical abstraction: It can show up in surprising ways and dramatically affect the properties of condensed matter, such as its ability to conduct electricity.

“These are three pioneers who are completely deserving of the honor,” says Charles Kane, a condensed matter theorist at the University of Pennsylvania (UPenn). The prize is divided with one half going to Thouless, and the other half split between Haldane and Kosterlitz. “When I heard the names I had to look to see which of their many contributions they were cited for,” says Eugene Mele, a theorist at UPenn.

In 1972, Kosterlitz and Thouless explained how topology can produce order where none is expected. The pair studied an abstract construct known as the XY model, which theorists had used to explain the onset of superfluidity in liquid helium—its ability to flow without resistance at temperatures below 2.17 K. That happens because all the atoms crowd into a single quantum wave that can be described by the XY model. However, in the 1960s theorists had argued that liquid helium in a 2D film could not become superfluid because thermal fluctuations would scramble the quantum wave.

Kosterlitz and Thouless showed otherwise. They realized that the lowest energy thermal fluctuations in the XY model would be tiny whirlpools called vortices and, at sufficiently low temperatures, they would appear only in countercirculating pairs with opposite topological charges. As a result, the effects of the paired vortices would largely cancel out, permitting a phase transition and superfluidity. In 1978, experimenters observed the predicted superfluidity in films of liquid helium.

In 1982, Thouless used topology to explain another odd phenomenon, the integer quantum Hall effect. It occurs when a thin semiconductor slab sits in a perpendicular magnetic field. As current runs along the slab, a voltage develops across it. The voltage increases with the strength of the magnetic field. But if the semiconductor is pure enough, the voltage goes up in specific jumps even as the magnetic field increases smoothly. Thouless showed that, at any given magnetic field, the complicated “bands” that describe the permitted energies and momenta of the electrons in a semiconductor have a specific topological charge. That charge can only change in jumps as the magnetic field increases, which explains the voltage steps.

In 1983, Haldane analyzed a purely abstract model: quantum mechanical particles arranged like pearls on a string. Each particle could interact with its neighbors, and its behavior would depend on its spin. Physicists use spin to divide particles into two classes: bosons, such as photons, and fermions, such as electrons.

Defying expectations, Haldane showed that strings of fermions and bosons behave very differently. A chain of fermions can be excited by an infinitesimally small energy perturbation. Fiddle with the chain at all, and ripples of torqueing spins zip up and down the line. On the other hand, a chain of bosons needs a good jolt for an excitation. The difference depends on the topology of the quantum wave describing the particles, and the result persuaded researchers that they could engineer material properties by controlling such topology.

Since then, the tools that the new Nobel laureates developed have become bedrock for the study of so-called topological phases of matter—perhaps the hottest field in condensed matter physics. For example, insulators such as bismuth selenide are known to become conductive on their surface because their twisted band structures unwind at the material’s edge. Other topological effects could lead to phenomena like unquenchable currents on the surfaces or edges of materials. Within materials, excitations that carry topological charges might even be braided with one another to perform quantum calculations. “A lot has been built on this foundation,” Kane says.

At a press conference announcing the prize, Haldane said that in the 1980s his work seemed very abstract. “I wrote in my paper that [my model] seemed to be an interesting toy model,” he said. Some toys are more interesting than others.

Those who had expected the prize to cite the year's blockbuster result, the detection of gravitational waves by the Laser Interferometer Gravitational-Wave Observatory (LIGO), will have to wait at least another year. Privately, some observers say they're happy that the Nobel Committee will have more time to think things over. Other prizes have lauded LIGO's founders: Rainer Weiss of the Massachusetts Institute of Technology in Cambridge and Ronald Drever and Kip Thorne of the California Institute of Technology (Caltech) in Pasadena. But some physicists say the award, which can be split at most three ways, should go to Caltech's Barry Barish, who directed construction of the massive machines in Louisiana and Washington state. Weiss, Drever, and Barish are in their 80s, so the committee may not have time to dither.