This is a chart of all
prime knots having seven or fewer
crossings (not including mirror images) along with the
unknot (or "
trivial knot"), a closed loop that is not a prime knot. The knots are labeled with
Alexander-Briggs notation. Many of these knots have special names, including the
trefoil knot (3
1) and
figure-eight knot (4
1).
Knot theory is the study of
knots viewed as different possible
embeddings of a
1-sphere (a
circle) in three-dimensional
Euclidean space (
R3). These mathematical objects are inspired by
real-world knots, such as knotted ropes or
shoelaces, but
don't have any free ends and so cannot be untied. (Two other closely related mathematical objects are
braids, which can have loose ends, and
links, in which two or more knots may be intertwined.) One way of distinguishing one knot from another is by the number of times its two-dimensional depiction crosses itself, leading to the numbering shown in the diagram above. The prime knots play a role very similar to
prime numbers in
number theory; in particular, any given (non-trivial) knot can be uniquely expressed as a "
sum" of prime knots (a series of prime knots spliced together) or is itself prime. Early knot theory enjoyed a brief period of popularity among physicists in the late 19th century after
William Thomson suggested that atoms are knots in the
luminiferous aether. This led to the first serious attempts to catalog all possible knots (which, along with links, now number in the billions). In the early 20th century, knot theory was recognized as a subdiscipline within
geometric topology. Scientific interest was resurrected in the latter half of the 20th century by the need to understand knotting problems in
organic chemistry, including the behavior of
DNA, and the recognition of connections between knot theory and
quantum field theory.