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 6₃ 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 6₂ knot. It is alternating,

hyperbolic 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 ...

, and fully amphichiral. It can be written as the

braid word A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...

\sigma_1^\sigma_2^2\sigma_1^\sigma_2. \,

Symmetry

Like the

figure-eight knot The figure-eight knot or figure-of-eight knot is a type of stopper knot. It is very important in both sailing and rock climbing as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under st ...

, the 6₃ knot is fully amphichiral. This means that the 6₃ knot is amphichiral, meaning that it is indistinguishable from its own mirror image. In addition, it is also

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 ...

, meaning that orienting the curve in either direction yields the same oriented knot.

Invariants

The

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 ...

of the 6₃ knot is :

\Delta(t) = t^2 - 3t + 5 - 3t^ + t^, \,

Conway polynomial is :

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

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^3 + 2q^2 - 2q + 3 - 2q^ + 2q^ - q^, \,

and the

Kauffman polynomial In knot theory, the Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. It is initially defined on a link diagram as :F(K)(a,z)=a^L(K)\,, where w(K) is the writhe of the link diagram and L(K) is a polynomial in ''a'' and ' ...

is :

L(a,z) = az^5 + z^5a^ + 2a^2z^4 + 2z^4a^ + 4z^4 + a^3z^3 + az^3 + z^3a^ + z^3a^ - 3a^2z^2 - 3z^2a^ - 6z^2 - a^3z - 2az - 2za^ - za^-3 + a^2 + a^ +3. \,

The 6₃ 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 5.69302.

References

{{DEFAULTSORT:6 3 knot Double torus knots and links