New! Sign up for our free email newsletter.
Reference Terms
from Wikipedia, the free encyclopedia

Knot theory

Knot theory is the mathematical branch of topology that studies mathematical knots, which are defined as embeddings of a circle in 3-dimensional Euclidean space, R3. This is basically equivalent to a conventional knotted string with the ends joined together to prevent it from becoming undone. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

Knots can be described in various ways. Given a method of description, however, there may be more than one description that represents the same knot. For example, a common method of describing a knot is a planar diagram. But any given knot can be drawn in many different ways using a planar diagram. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot. One way of distinguishing knots is by using a knot invariant, a "quantity" which remains the same even with different descriptions of a knot.

The concept of a knot has been extended to higher dimensions by considering n-dimensional spheres in m-dimensional Euclidean space. This was investigated most actively in the period 1960-1980, when a number of breakthroughs were made. In recent years, low dimensional phenomena have garnered the most interest.

Research in knot theory began with the creation of knot tables and the systematic tabulation of knots. While tabulation remains an important task, today's researchers have a wide variety of backgrounds and goals.

In the last 30 years, knot theory has also become a tool in applied mathematics. Chemists and biologists use knot theory to understand, for example, chirality of molecules and the actions of enzymes on DNA.

Related Stories
 


Matter & Energy News

October 29, 2025

Tohoku University researchers have found a way to make quantum sensors more sensitive by connecting superconducting qubits in optimized network patterns. These networks amplify faint signals possibly left by dark matter. The approach outperformed ...
A UCLA-led team has achieved the sharpest-ever view of a distant star’s disk using a groundbreaking photonic lantern device on a single telescope—no multi-telescope array required. This technology splits incoming starlight into multiple ...
Researchers propose that hydrogen gas from the early Universe emitted detectable radio waves influenced by dark matter. Studying these signals, especially from the Moon’s radio-quiet environment, could reveal how dark matter clumped together ...
A team of researchers has designed a theoretical model for a topological quantum battery capable of long-distance energy transfer and immunity to dissipation. By exploiting topological properties in photonic waveguides, they showed that energy loss ...
Scientists have developed a chromium-molybdenum-silicon alloy that withstands extreme heat while remaining ductile and oxidation-resistant. It could replace nickel-based superalloys, which are limited to about 1,100°C. The new material might make ...
Researchers have found that 2D materials can self-form microscopic cavities that trap light and electrons, altering their quantum behavior. With a miniaturized terahertz spectroscope, the team observed standing light-matter waves without needing ...
A new light-driven cancer therapy uses LEDs and tin nanoflakes to kill tumors safely and affordably. Developed by teams in Texas and Portugal, it eliminates up to 92% of skin cancer cells without ...
Researchers from NTNU and EPFL have unveiled a compact, low-cost laser that outperforms current models in speed, control, and precision. Built using microchip technology, it can be mass-produced for use in everything from Lidar navigation to gas ...
A collaboration between the University of Michigan and AFRL has resulted in 3D-printed metamaterials that can block vibrations using complex geometries. Inspired by nature and theoretical physics, these “kagome tubes” demonstrate how geometry ...
Researchers discovered how to stabilize a high-performance sodium compound, giving sodium-based solid-state batteries the power and stability they’ve long lacked. The new material conducts ions far ...
A team of engineers at North Carolina State University has designed a polymer “Chinese lantern” that can rapidly snap into multiple stable 3D shapes—including a lantern, a spinning top, and more—by compression or twisting. By adding a ...
Scientists have developed an ultra-thin, paper-like LED that emits a warm, sunlike glow, promising to revolutionize how we light up our homes, devices, and workplaces. By engineering a balance of red, yellow-green, and blue quantum dots, the ...

Latest Headlines

updated 12:56 pm ET