6₃ knot
Encyclopedia
In knot theory
, the 63 knot is one of three prime knot
s 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. It is also invertible
, meaning that orienting the curve in either direction yields the same oriented knot.
The Alexander polynomial
of the 63 knot is
the Conway polynomial
is
and the Jones polynomial is
The 63 knot is a hyperbolic knot, with its complement having a volume
of approximately 5.69302.
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 63 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 stevedore knot
Stevedore knot (mathematics)
In knot theory, the stevedore knot is one of three prime knots 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, and it can also be described as a twist knot with four twists, or as the pretzel...
and 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...
.
Like the figure-eight knot
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...
, the 63 knot is 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...
, meaning that it is indistinguishable from its own mirror image. It is also 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...
, meaning that orienting the curve in either direction yields the same oriented knot.
The 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...
of the 63 knot is
the Conway polynomial
Conway polynomial
In mathematics, Conway polynomial can refer to:* the Alexander–Conway polynomial in knot theory* the Conway polynomial...
is
and the Jones polynomial is
The 63 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 5.69302.