Knot theory is the study of knots and links.

Roughly a knot is a simple closed curve in $\mathbb{R}^{3}$, and two knots are considered equivalent if and only if one can be smoothly deformed into another. This will often be used as a working definition as it is simple and appeals to intuition. Unfortunately this definition can not be taken too seriously because it includes many pathological cases, or wild knots, such as the connected sum of an infinite number of trefoils. Furthermore one must be careful about defining a “smooth deformation”, or all knots might turn out to be equivalent! (We shouldn’t be allowed to shrink part of a knot down to nothing.)

Links are defined in terms of knots, so once we have a definition for knots we have no trouble defining them.

Each knot is a component of the link. In particular a knot is a link of one component.

Luckily the knot theorist is not usually interested in the exact form of a knot or link, but rather the in its equivalence class. (Even so a possible formal definition for knots is given at the end of this entry.) All the “interesting” information about a knot or link can be described using a knot diagram. (It should be noted that the words “knot” and “link” are often used to mean an equivalence class of knots or links respectively. It is normally clear from context if this usage is intended.)

A knot diagram is a projection of a link onto a plane such that no more than two points of the link are projected to the same point on the plane and at each such point it is indicated which strand is closest to the plane (usually by erasing part of the lower strand). This can best be explained with some examples:

some knot diagrams

Two different knot diagrams may both represent the same knot — for example the last two diagrams both represent the unknot, although this is not obvious. Much of knot theory is devoted to telling when two knot diagrams represent the same link. In one sense this problem is easy — two knot diagrams represent equivalent links if and only if there exists a sequence of Reidemeister moves transforming one diagram to another.

Note that number 1. is the inverse of number 2. and number 3. is the inverse of number 4. Number 5 is its own inverse. In pictures:

$\longleftrightarrow$
   $\longleftrightarrow$   
   $\longleftrightarrow$   

Finding such a sequence of Reidemeister moves is generally not easy, and proving that no such sequence exists can be very difficult, so other approaches must be taken.

Knot theorists have accumulated a large number of knot invariants, values associated with a knot diagram which are unchanged when the diagram is modified by a Reidemeister move. Two diagrams with the same invariant may not represent the same knot, but two diagrams with different invariant never represent the same knot.

Knot theorists also study ways in which a complex knot may be described in terms of simple pieces — for example every knot is the connected sum of non trivial prime knots and many knots can be described simply using Conway notation.