knot theory In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...

, the stevedore knot is one of three

prime knot In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be co ...

s with crossing number six, the others being the 6₂ knot and the 6₃ knot. The stevedore knot is listed as the 6₁ knot in the

Alexander–Briggs notation In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ...

, and it can also be described as a

twist knot In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together. (That is, a twist knot is any Whitehead double of an unknot.) The twist knots are an infinite f ...

with four twists, or as the (5,−1,−1)

pretzel knot A Pretzel knot may refer to: * Pretzel link: a concept in mathematics * Soft pretzel with garlic * Stafford knot The Stafford knot, more commonly known as the Staffordshire knot, is a distinctive three-looped knot that is the traditional symbol o ...

. The mathematical stevedore knot is named after the common

stevedore knot The stevedore knot is a stopper knot, often tied near the end of a rope. It is more bulky and less prone to jamming than the closely related figure-eight knot. Naming There is a lack of consensus among knot experts regarding the origin of t ...

, which is often used as a

stopper Stopper may refer to: * Bung, a plug used to stop the opening of a container ** Laboratory rubber stopper, a specific type of bung * Plug (sanitation), used to stop a drainage outlet * Defender (association football), in soccer (association footba ...

at the end of a

rope A rope is a group of yarns, plies, fibres, or strands that are twisted or braided together into a larger and stronger form. Ropes have tensile strength and so can be used for dragging and lifting. Rope is thicker and stronger than similarly ...

. The mathematical version of the knot can be obtained from the common version by joining together the two loose ends of the rope, forming a knotted

loop Loop or LOOP may refer to: Brands and enterprises * Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live * Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets * Loop Mobile, an ...

. The stevedore knot is

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...

but not amphichiral. Its

Alexander polynomial In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a ve ...

is :

\Delta(t) = -2t+5-2t^, \,

its Conway polynomial is :

\nabla(z) = 1-2z^2, \,

and its

Jones polynomial In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynom ...

is :

V(q) = q^2-q+2-2q^+q^-q^+q^. \,

The Alexander polynomial and Conway polynomial are the same as those for the knot 9₄₆, but the Jones polynomials for these two knots are different. Because the Alexander polynomial is not monic, the stevedore knot is not fibered. The stevedore knot is a

ribbon knot In the mathematical area of knot theory, a ribbon knot is a knot that bounds a self-intersecting disk with only ''ribbon singularities''. Intuitively, this kind of singularity can be formed by cutting a slit in the disk and passing another part of ...

, and is therefore also a

slice knot A slice knot is a mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space. Definition A knot K \subset S^3 is said to be a topologically or smoothly slice knot, if it is the boundary of an embedded disk in ...

. The stevedore knot is a

hyperbolic knot Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

, with its complement having a

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

of approximately 3.16396.

References

{{Knot theory, state=collapsed Double torus knots and links

See also

References