In the mathematical field of knot theory, the tricolorability of a knot 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 is not tricolorable, any tricolorable knot is necessarily nontrivial.
In these rules a strand in a knot diagram will be a piece of the string that goes from one undercrossing to the next. [1] A knot is tricolorable if each strand of the knot diagram can be colored one of three colors, subject to the following rules: [2]
Some references state instead that all three colors must be used. [3] 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 are not. If the projection of a knot is tricolorable, then Reidemeister moves on the knot preserve tricolorability, so either every projection of a knot is tricolorable or none is." [2]
Here is an example of how to color a knot in accordance of the rules of tricolorability. By convention, knot theorists use the colors red, green, and blue.
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 knots all red would also give an admissible coloring. The true lover's knot is also tricolorable. [4]
Tricolorable knots with less than nine crossings include 61, 74, 77, 85, 810, 811, 815, 818, 819, 820, and 821.
The figure-eight knot 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.
Tricolorability is an isotopy invariant, which is a property of a knot or link that remains constant regardless of any ambient isotopy. This can be proven by examining Reidemeister moves. Since each Reidemeister move can be made without affecting tricolorability, tricolorability is an isotopy invariant.
| Reidemeister Move I is tricolorable. | Reidemeister Move II is tricolorable. | Reidemeister Move III is tricolorable. |
|---|---|---|
 |  |  |
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.
If the torus knot/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.
| Hyperbolic |
|
|---|---|
| Satellite | |
| Torus | |
| Invariants | |
| Notation and operations | |
| Other | |
In the 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 tied into it, unknotted. To a knot theorist, an unknot is any embedded topological circle in the 3-sphere that is ambient isotopic to a geometrically round circle, the standard unknot.
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 knot being a ring. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.
In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers (algebraic), but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory (for example, "a knot invariant is a rule that assigns to any knot K a quantity φ(K) such that if K and K' are equivalent then φ(K) = φ(K')."). Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics. Knot invariants are thus used in knot classification, both in "enumeration" and "duplication removal".
A knot invariant is a quantity defined on the set of all knots, which takes the same value for any two equivalent knots. For example, a knot group is a knot invariant.
Typically a knot invariant is a combinatorial quantity defined on knot diagrams. Thus if two knot diagrams differ with respect to some knot invariant, they must represent different knots. However, as is generally the case with topological invariants, if two knot diagrams share the same values with respect to a [single] knot invariant, then we still cannot conclude that the knots are the same.
Skein relations are a mathematical tool used to study knots. A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One way to answer the question is using knot polynomials, which are invariants of the knot. If two diagrams have different polynomials, they represent different knots. In general, the converse does not hold.
In mathematics, a knot is an embedding of the circle S1 into three-dimensional Euclidean space, R3. Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of R3 which takes one knot to the other.
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory.
In knot theory, there are several competing notions of the quantity writhe, or . In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amount of "coiling" of a mathematical knot in three-dimensional space and assumes real numbers as values. In both cases, writhe is a geometric quantity, meaning that while deforming a curve in such a way that does not change its topology, one may still change its writhe.
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 polynomial in the variable with integer coefficients.
In the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram. Kurt Reidemeister (1927) and, independently, James Waddell Alexander and Garland Baird Briggs (1926), demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a sequence of the three Reidemeister moves.
In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a trivial reference link, usually called the unlink, but the word is also sometimes used in context where there is no notion of a trivial link.
In mathematics, a Seifert surface is an orientable surface whose boundary is a given knot or link.
In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf.
In the mathematical field of knot theory, the bracket polynomial is a polynomial invariant of framed links. Although it is not an invariant of knots or links, a suitably "normalized" version yields the famous knot invariant called the Jones polynomial. The bracket polynomial plays an important role in unifying the Jones polynomial with other quantum invariants. In particular, Kauffman's interpretation of the Jones polynomial allows generalization to invariants of 3-manifolds.
In the mathematical field of knot theory, Fox n-coloring is a method of specifying a representation of a knot group or a group of a link onto the dihedral group of order n where n is an odd integer by coloring arcs in a link diagram. Ralph Fox discovered this method "in an effort to make the subject accessible to everyone" when he was explaining knot theory to undergraduate students at Haverford College in 1956. Fox n-coloring is an example of a conjugation quandle.
In the mathematical subject of knot theory, regular isotopy is the equivalence relation of link diagrams that is generated by using the 2nd and 3rd Reidemeister moves only. The notion of regular isotopy was introduced by Louis Kauffman. It can be thought of as an isotopy of a ribbon pressed flat against the plane which keeps the ribbon flat. For diagrams in the plane this is a finer equivalence relation than ambient isotopy of framed links, since the 2nd and 3rd Reidemeister moves preserve the winding number of the diagram. However, for diagrams in the sphere, the two notions are equivalent, due to the extra freedom of passing a strand through infinity.
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
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. The twist knots are an infinite family of knots, and are considered the simplest type of knots after the torus knots.
In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself to untie it. If a knot has unknotting number , then there exists a diagram of the knot which can be changed to unknot by switching crossings. The unknotting number of a knot is always less than half of its crossing number. This invariant was first defined by Hilmar Wendt in 1936.
Knots Unravelled: From String to Mathematics is a book on the mathematics of knots, intended for schoolchildren and other non-mathematicians. It was written by mathematician Meike Akveld and mathematics publisher Andrew Jobbings, and published in 2011 by Arbelos, Jobbings's firm.
