Square knot sum of trefoils stick number

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 square knot is a composite knot obtained by taking 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 classifi ...

of a

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 ...

with its reflection. It is closely related to the

granny knot The granny knot is a binding knot, used to secure a rope or line around an object. It is considered inferior to the reef knot (square knot), which it superficially resembles. Neither of these knots should be used as a bend knot for attaching t ...

, which is also a connected sum of two trefoils. Because the trefoil knot is the simplest nontrivial knot, the square knot and the granny knot are the simplest of all composite knots. The square knot is the mathematical version of the common

reef knot The reef knot, or square knot, is an ancient and simple binding knot used to secure a rope or line around an object. It is sometimes also referred to as a Hercules knot or Heracles knot. The knot is formed by tying a left-handed overhand knot ...

Construction

The square knot can be constructed from two trefoil knots, one of which must be left-handed and the other right-handed. Each of the two knots is cut, and then the loose ends are joined together pairwise. The resulting connected sum is the square knot. It is important that the original trefoil knots be mirror images of one another. If two identical trefoil knots are used instead, the result is a granny knot.

Properties

The square knot is amphichiral, meaning that it is indistinguishable from its own mirror image. The crossing number of a square knot is six, which is the smallest possible crossing number for a composite knot. 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 ...

of the square knot is :

\Delta(t) = (t - 1 + t^)^2, \,

which is simply the

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

of the Alexander polynomial of a trefoil knot. Similarly, the

Alexander–Conway 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 ...

of a square knot is :

\nabla(z) = (z^2+1)^2.

These two polynomials are the same as those for the granny knot. However, the

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 polyno ...

for the square knot is :

V(q) = (q^ + q^ - q^)(q + q^3 - q^4) = -q^3 + q^2 - q + 3 - q^ + q^ - q^. \,

This is the product of the Jones polynomials for the right-handed and left-handed trefoil knots, and is different from the Jones polynomial for a granny knot. The knot group of the square knot is given by the presentation :

\langle x, y, z \mid x y x = y x y, x z x = z x z \rangle. \,

This is

isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

to the knot group of the granny knot, and is the simplest example of two different knots with isomorphic knot groups. Unlike the granny knot, the square knot is a

ribbon knot In the mathematical area of knot theory, a ribbon knot is a knot that bounds a self-intersecting disk with only ''ribbon singularities''. Intuitively, this kind of singularity can be formed by cutting a slit in the disk and passing another part o ...

, and it is therefore also a

slice knot A slice knot is a knot (mathematics), mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space. Definition A knot K \subset S^3 is said to be a topologically slice knot or a smoothly slice knot, if it is the ...

References

{{Knot theory, state=collapsed