This article contains a list of
mathematical knots and
links. See also
list of knots
This list of knots includes many alternative names for common knots and lashings. Knot names have evolved over time, and there are many conflicting or confusing naming issues. The overhand knot, for example, is also known as the thumb knot. The ...
,
list of geometric topology topics.
Knots
Prime knots
*0
1 knot/
Unknot
In the knot theory, mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a Knot (mathematics), knot tied into it, unknotted. To a knot ...
- a simple un-knotted closed loop
*3
1 knot/
Trefoil knot
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot (mathematics), knot. The trefoil can be obtained by joining the two loose ends of a common overhand knot, resulting in a knotted loop (topology ...
- (2,3)-torus knot, the two loose ends of a common overhand knot joined together
*4
1 knot/
Figure-eight knot (mathematics)
In knot theory, a figure-eight knot (also called Listing's knot) is the unique knot with a crossing number (knot theory), crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the unknot and th ...
- a prime knot with a crossing number four
*5
1 knot/
Cinquefoil knot, (5,2)-torus knot, Solomon's seal knot, pentafoil knot - a prime knot with crossing number five which can be arranged as a star polygon (
pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around ...
)
*5
2 knot/
Three-twist knot - the twist knot with three-half twists
*6
1 knot/
Stevedore knot (mathematics) - a prime knot with crossing number six, it can also be described as a twist knot with four twists
*
62 knot - a prime knot with crossing number six
*
63 knot - a prime knot with crossing number six
*
71 knot, septafoil knot, (7,2)-torus knot - a prime knot with crossing number seven, which can be arranged as a star polygon (
heptagram
A heptagram, septagram, septegram or septogram is a seven-point star polygon, star drawn with seven straight strokes.
The name ''heptagram'' combines a numeral prefix, ''hepta-'', with the Greek language, Greek suffix ''wikt:-gram, -gram ...
)
*
74 knot, "endless knot"
*
818 knot, "carrick mat"
*10
161/10
162, known as the
Perko pair; this was a single knot listed twice in Dale Rolfsen's knot table; the duplication was discovered by Kenneth Perko
*12n242/
(−2,3,7) pretzel knot
*(''p'', ''q'')-
torus knot
In knot theory, a torus knot is a special kind of knot (mathematics), knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link (knot theory), link which lies on the surface of a torus in the same way. Each t ...
- a special kind of knot that lies on the surface of an unknotted torus in R
3
Composite
*
Square knot (mathematics) - a composite knot obtained by taking the connected sum of a trefoil knot with its reflection
*
Granny knot (mathematics) - a composite knot obtained by taking the connected sum of two identical trefoil knots
Links
*0 link/
Unlink - equivalent under ambient isotopy to finitely many disjoint circles in the plane
*2 link/
Hopf link
In mathematics, mathematical knot theory, the Hopf link is the simplest nontrivial link (knot theory), link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf.
Geometric realizat ...
- the simplest nontrivial link with more than one component; it consists of two circles linked together exactly once (L2a1)
*4 link/
Solomon's knot (a two component "link" rather than a one component "knot") - a traditional decorative motif used since ancient times (L4a1)
*5 link/
Whitehead link - two projections of the unknot: one circular loop and one figure eight-shaped loop intertwined such that they are inseparable and neither loses its form (L5a1)
*
Brunnian link - a nontrivial link that becomes trivial if any component is removed
*6 link/
Borromean rings
In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are link (knot theory), topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops wh ...
- three topological circles which are linked and form a Brunnian link (L6a4)
*
L10a140 link - presumably the simplest non-Borromean Brunnian link
*
Pretzel link - a
Montesinos link with integer tangles
External links
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{{Knot theory
Knots and links
Knot theory