In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

knot theory In 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 joined so it cannot be und ...

, the tricolorability of a

knot A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ...

is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an isotopy invariant, and hence can be used to distinguish between two different (non- isotopic) knots. In particular, since the

unknot In the knot theory, mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a Knot (mathematics), knot tied into it, unknotted. To a knot ...

is not tricolorable, any tricolorable knot is necessarily nontrivial.

Rules of tricolorability

In these rules a strand in a

knot diagram In 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 be undone, the simplest k ...

will be a piece of the string that goes from one undercrossing to the next. A knot is tricolorable if each strand of the knot diagram can be colored one of three colors, subject to the following rules:Weisstein, Eric W. (2010). ''CRC Concise Encyclopedia of Mathematics'', Second Edition, p.3045. . quoted at Accessed: May 5, 2013. :1. At least two colors must be used, and :2. At each crossing, the three incident strands are either all the same color or all different colors. Some references state instead that all three colors must be used.Gilbert, N.D. and Porter, T. (1994) ''Knots and Surfaces'', p. 8 For a knot, this is equivalent to the definition above; however, for a link it is not. "The trefoil knot and trivial 2-link are tricolorable, but the unknot, Whitehead link, and

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

are not. If the projection of a knot is tricolorable, then

Reidemeister moves In the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram. and, independently, , demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a seque ...

on the knot preserve tricolorability, so either every projection of a knot is tricolorable or none is."

Examples

Here is an example of how to

color Color (or colour in English in the Commonwealth of Nations, Commonwealth English; American and British English spelling differences#-our, -or, see spelling differences) is the visual perception based on the electromagnetic spectrum. Though co ...

a knot in accordance of the rules of tricolorability. By convention, knot theorists use the colors red, green, and blue.

Example of a tricolorable knot

The granny knot is tricolorable. In this coloring the three strands at every crossing have three different colors. Coloring one but not both of the

trefoil knot In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot (mathematics), knot. The trefoil can be obtained by joining the two loose ends of a common overhand knot, resulting in a knotted loop (topology ...

s all red would also give an admissible coloring. The true lover's knot is also tricolorable. Tricolorable knots with less than nine crossings include 6₁, 7₄, 7₇, 8₅, 8₁₀, 8₁₁, 8₁₅, 8₁₈, 8₁₉, 8₂₀, and 8₂₁.

Example of a non-tricolorable knot

The

is not tricolorable. In the diagram shown, it has four strands with each pair of strands meeting at some crossing. If three of the strands had the same color, then all strands would be forced to be the same color. Otherwise each of these four strands must have a distinct color. Since tricolorability is a knot invariant, none of its other diagrams can be tricolored either.

Isotopy invariant

Tricolorability is an isotopy invariant, which is a property of a knot or link that remains constant regardless of any

ambient isotopy In the mathematical subject of topology, an ambient isotopy, also called an ''h-isotopy'', is a kind of continuous distortion of an ambient space, for example a manifold, taking a submanifold to another submanifold. For example in knot theory, o ...

. This can be proven for tame knots by examining Reidemeister moves. Since each Reidemeister move can be made without affecting tricolorability, tricolorability is an isotopy invariant of tame knots.

Properties

Because tricolorability is a binary classification (a link is either tricolorable or not*), it is a relatively weak invariant. The composition of a tricolorable knot with another knot is always tricolorable. A way to strengthen the invariant is to count the number of possible 3-colorings. In this case, the rule that at least two colors are used is relaxed and now every link has at least three 3-colorings (just color every arc the same color). In this case, a link is 3-colorable if it has more than three 3-colorings. Any separable link with a tricolorable separable component is also tricolorable.

In torus knots

If the

torus knot In knot theory, a torus knot is a special kind of knot (mathematics), knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link (knot theory), link which lies on the surface of a torus in the same way. Each t ...

/link denoted by (m,n) is tricolorable, then so are (j*m,i*n) and (i*n,j*m) for any natural numbers i and j.