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 6₂ 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

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 ...

and the 6₃ knot. This knot is sometimes referred to as the Miller Institute knot, because it appears in the logo of the

Miller Institute The Miller Institute for Basic Research in Science was established on the University of California, Berkeley, campus in 1955 after Adolph C. Miller and his wife, Mary Sprague Miller, made a donation to the university. It was their wish that the d ...

for Basic Research in Science at the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...

. The 6₂ 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) = -t^2 + 3t -3 + 3t^ - t^, \,

its Conway polynomial is :

\nabla(z) = -z^4 - z^2 + 1, \,

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 - 1 + 2q^ - 2q^ + 2q^ - 2q^ + q^. \,

The 6₂ 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 A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...

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 4.40083.

Surface

File:Superfície - bordo Nó 6,2.jpg, Surface of knot 6.2

Example

Ways to assemble of knot 6.2 File:6₂ knot.webm, Example 1 File:6₂ knot (2).webm, Example 2 If a

bowline The bowline ( or ) is an ancient and simple knot used to form a fixed loop at the end of a rope. It has the virtues of being both easy to tie and untie; most notably, it is easy to untie after being subjected to a load. The bowline is sometimes ...

is tied and the two free ends of the rope are brought together in the simplest way, the knot obtained is the 6₂ knot. The sequence of necessary moves are depicted here: Image:Bowline to 6 2 knot.gif, From a bowline (ends connected) to the 6₂ knot.

References

{{DEFAULTSORT:6 2 knot Double torus knots and links