List of mathematical knots and links

Knots

0₁ knot/Unknot - a simple un-knotted closed loop
3₁ knot/Trefoil knot - (2,3)-torus knot, the two loose ends of a common overhand knot joined together
4₁ knot/Figure-eight knot (mathematics) - a prime knot with a crossing number four
5₁ 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 {5/2} star polygon (pentagram)
5₂ knot/Three-twist knot - the twist knot with three-half twists
6₁ 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 {7/2} star polygon (heptagram)
7₄ knot, "endless knot"
8₁₈ knot, "carrick mat"
10₁₆₁/10₁₆₂, 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 - a special kind of knot that lies on the surface of an unknotted torus in R³

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

0²
₁ link/Unlink - equivalent under ambient isotopy to finitely many disjoint circles in the plane
2²
₁ link/Hopf link - the simplest nontrivial link with more than one component; it consists of two circles linked together exactly once (L2a1)
4²
₁ link/Solomon's knot (a two component "link" rather than a one component "knot") - a traditional decorative motif used since ancient times (L4a1)
5²
₁ 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
6³
₂ link/Borromean rings - 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