To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

# Granny knot (mathematics)

3D depiction

In knot theory, the granny knot is a composite knot obtained by taking the connected sum of two identical trefoil knots. It is closely related to the square knot, which can also be described as a connected sum of two trefoils. Because the trefoil knot is the simplest nontrivial knot, the granny knot and the square knot are the simplest of all composite knots.

The granny knot is the mathematical version of the common granny knot.

• 1/3
Views:
30 251 076
7 566
1 923
• 10 more amazing bets you will always win (5)
• Flexagon Cushions
• Clove Hitch knot how to - Κόμπος Ψαλιδιά Πώς την κάνουμε

## Construction

The granny knot can be constructed from two identical trefoil knots, which must either be both left-handed or both right-handed. Each of the two knots is cut, and then the loose ends are joined together pairwise. The resulting connected sum is the granny knot.

It is important that the original trefoil knots be identical to each another. If mirror-image trefoil knots are used instead, the result is a square knot.

## Properties

The crossing number of a granny knot is six, which is the smallest possible crossing number for a composite knot. Unlike the square knot, the granny knot is not a ribbon knot or a slice knot.

The Alexander polynomial of the granny knot is

${\displaystyle \Delta (t)=(t-1+t^{-1})^{2},}$

which is simply the square of the Alexander polynomial of a trefoil knot. Similarly, the Conway polynomial of a granny knot is

${\displaystyle \nabla (z)=(z^{2}+1)^{2}.}$

These two polynomials are the same as those for the square knot. However, the Jones polynomial for the (right-handed) granny knot is

${\displaystyle V(q)=(q^{-1}+q^{-3}-q^{-4})^{2}=q^{-2}+2q^{-4}-2q^{-5}+q^{-6}-2q^{-7}+q^{-8}.}$

This is the square of the Jones polynomial for the right-handed trefoil knot, and is different from the Jones polynomial for a square knot.

The knot group of the granny knot is given by the presentation

${\displaystyle \langle x,y,z\mid xyx=yxy,xzx=zxz\rangle .}$[1]

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

## References

1. ^ Weisstein, Eric W. "Granny Knot". MathWorld.