In knot theory

Knot theory

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...

, the stevedore knot is one of three prime knot

Prime knot

In knot theory, a prime knot 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 composite. It can be a nontrivial problem to determine whether a...

s with crossing number

Crossing number (knot theory)

In the mathematical area of knot theory, the crossing number of a knot is the minimal number of crossings of any diagram of the knot. It is a knot invariant....

 six, the others being the 6₂ knot

6₂ knot

In knot theory, the 62 knot is one of three prime knots with crossing number six, the others being the stevedore knot and the 63 knot. This knot is sometimes referred to as the Miller Institute knot, because is appears in the logo of the Miller Institute for Basic Research in Science at the...

 and the 6₃ knot

6₃ knot

In knot theory, the 63 knot is one of three prime knots with crossing number six, the others being the stevedore knot and the 62 knot.Like the figure-eight knot, the 63 knot is amphichiral, meaning that it is indistinguishable from its own mirror image...

. The stevedore knot is listed as the 6₁ knot in the Alexander–Briggs notation, and it can also be described as a twist knot

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...

 with four twists, or as the (5,−1,−1) pretzel knot.

The mathematical stevedore knot is named after the common stevedore knot, which is often used as a stopper

Stopper (knot)

The term stopper knot has three distinct meanings in the context of knotting and cordage.-At the end of a line:A stopper knot is tied at the end of a rope to prevent the end from unraveling, slipping through another knot, or passing back through a hole, block, or belay/rappel device...

 at the end of a rope

Rope

A rope is a length of fibres, twisted or braided together to improve strength for pulling and connecting. It has tensile strength but is too flexible to provide compressive strength...

. 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 (topology)

In mathematics, a loop in a topological space X is a path f from the unit interval I = [0,1] to X such that f = f...

.

The stevedore knot is invertible

Invertible knot

In mathematics, especially in the area of topology known as knot theory, an invertible knot is a knot that can be continuously deformed to itself, but with its orientation reversed. A non-invertible knot is any knot which does not have this property. The invertibility of a knot is a knot invariant...

 but not amphichiral

Amphichiral knot

In the mathematical field of knot theory, a chiral knot is a knot that is not equivalent to its mirror image. An oriented knot that is equivalent to its mirror image is an amphichiral knot, also called an achiral knot or amphicheiral knot. The chirality of a knot is a knot invariant...

. Its Alexander polynomial

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...

 is


its Conway polynomial

Conway polynomial

In mathematics, Conway polynomial can refer to:* the Alexander–Conway polynomial in knot theory* the Conway polynomial...

 is


and its Jones polynomial is


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

Fibered knot

A knot or link Kin the 3-dimensional sphere S^3 is called fibered or fibred if there is a 1-parameter family F_t of Seifert surfaces for K, where the parameter t runs through the points of the unit circle S^1, such that if s is not equal to t...

.

The stevedore knot is a ribbon knot

Ribbon knot

In the mathematical area of knot theory, a ribbon knot is a knot that bounds a self-intersecting disc with only ribbon singularities. This type of singularity is a self-intersection along an arc; the preimage of this arc consists of two arcs in the disc, one properly embedded in the disc and the...

, and is therefore also a slice knot.

The stevedore knot is a hyperbolic knot, with its complement having a volume

Hyperbolic volume (knot)

In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is simply the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily finite. The hyperbolic volume of a non-hyperbolic knot is often defined to be zero...

 of approximately 3.16396.

See also

• Figure-eight knot (mathematics)
  Figure-eight knot (mathematics)
  In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot and trefoil knot...
  
  
• Knot theory
  Knot theory
  In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...