In the mathematical theory of knots, a pretzel link is a special kind of link. It consists of a finite number tangles made of two intertwined circular helices. The tangles are connected cyclicly, the first component of the first tangle is connected to the second component of the second tangle, etc., with the first component of the last tangle connected to the second component of the first. A pretzel link which is also 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, ...

(i.e. a link with one component) is a pretzel knot. Each tangle is characterized by its number of twists, positive if they are counter-clockwise or left-handed, negative if clockwise or right-handed. In the standard projection of the

(p_1,\,p_2,\dots,\,p_n)

pretzel link, there are

p_1

left-handed crossings in the first , tangle,

p_2

in the second, and, in general,

p_n

in the nth. A pretzel link can also be described as a Montesinos link with integer tangles.

Some basic results

The

(p_1,p_2,\dots,p_n)

pretzel link is a

iff In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

both

n

and all the

p_i

are odd or exactly one of the

p_i

is even.Kawauchi, Akio (1996). ''A survey of knot theory''. Birkhäuser. The

(p_1,\,p_2,\dots,\,p_n)

pretzel link is

split Split(s) or The Split may refer to: Places * Split, Croatia, the largest coastal city in Croatia * Split Island, Canada, an island in the Hudson Bay * Split Island, Falkland Islands * Split Island, Fiji, better known as Hạfliua Arts, entertain ...

if at least two of the

p_i

are

zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...

; but the converse is false. The

(-p_1,-p_2,\dots,-p_n)

pretzel link is the

mirror image A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off from substances ...

of the

(p_1,\,p_2,\dots,\,p_n)

pretzel link. The

(p_1,\,p_2,\dots,\,p_n)

pretzel link is isotopic to the

(p_2,\,p_3,\dots,\,p_n,\,p_1)

pretzel link. Thus, too, the

(p_1,\,p_2,\dots,\,p_n)

pretzel link is isotopic to the

(p_k,\,p_,\dots,\,p_n,\,p_1,\,p_2,\dots,\,p_)

pretzel link. The

(p_1,\,p_2,\,\dots,\,p_n)

pretzel link is isotopic to the

(p_n,\,p_,\dots,\,p_2,\,p_1)

pretzel link. However, if one orients the links in a canonical way, then these two links have opposite orientations.

Some examples

The (1, 1, 1) pretzel knot is the (right-handed)

trefoil A trefoil () is a graphic form composed of the outline of three overlapping rings, used in architecture and Christian symbolism, among other areas. The term is also applied to other symbols with a threefold shape. A similar shape with four ring ...

; the (−1, −1, −1) pretzel knot is its

. The (5, −1, −1) pretzel knot is 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 ...

(6₁). If p, q, r are distinct odd integers greater than 1, then the (''p'', ''q'', ''r'') pretzel knot is a non-invertible knot. The (2''p'', 2''q'', 2''r'') pretzel link is a link formed by three linked

unknot 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 embe ...

s. The (−3, 0, −3) pretzel knot ( square knot (mathematics)) is the

connected sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classific ...

of two

trefoil knot 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 together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest kn ...

s. The (0, q, 0) pretzel link is the split union of an

and another knot.

Montesinos

A Montesinos link is a special kind of link that generalizes pretzel links (a pretzel link can also be described as a Montesinos link with integer tangles). A Montesinos link which is also a

(i.e., a link with one component) is a Montesinos knot. A Montesinos link is composed of several rational tangles. One notation for a Montesinos link is

K(e;\alpha_1 /\beta_1,\alpha_2  /\beta_2,\ldots,\alpha_n  /\beta_n)

. In this notation,

e

and all the

\alpha_i

and

\beta_i

are integers. The Montesinos link given by this notation consists of the

sum Sum most commonly means the total of two or more numbers added together; see addition. Sum can also refer to: Mathematics * Sum (category theory), the generic concept of summation in mathematics * Sum, the result of summation, the additio ...

of the rational tangles given by the integer

e

and the rational tangles

\alpha_1 /\beta_1,\alpha_2  /\beta_2,\ldots,\alpha_n  /\beta_n

These knots and links are named after the Spanish topologist José María Montesinos Amilibia, who first introduced them in 1973.

Utility

(−2, 3, 2n + 1) pretzel links are especially useful in the study of

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds ...

s. Many results have been stated about the manifolds that result from

Dehn surgery In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: ''drilling'' then '' ...

on the (−2,3,7) pretzel knot in particular. The

hyperbolic volume In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily a finite real number, and is a topological inva ...

of the complement of the pretzel link is times

Catalan's constant In mathematics, Catalan's constant , is defined by : G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots, where is the Dirichlet beta function. Its numerical value is approximately : It is not known whether is irra ...

, approximately 3.66. This pretzel link complement is one of two two-cusped hyperbolic manifolds with the minimum possible volume, the other being the complement of the

Whitehead link In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links. It can be drawn as an alternating link with five crossings, from the overlay of a circle and a figure-eight shaped loop. Structure A common way ...

Some basic results

Some examples

Montesinos

Utility

References

Further reading