The granny knot is a composite knot of six crossings consisting of a knot sum of two left-handed trefoils
with the same orientation (Rolfsen 1976, p. 220).
The granny knot has the same Alexander polynomial as the square
knot. It also has the same group as the square knot
(Rolfsen 1976, p. 62). However, the two knots are distinct, and Fox (1952) showed
that the knot complements of the square and granny
knots are not homeomorphic (Rolfsen 1976, p. 62).
In addition, the square knot is a slice
knot, while the granny knot is not (Rolfsen 1976, p. 220).
Cipra, B. A. "To Have and Have Knot: When Are Knots Alike?" Science241, 1291-1292, 1988.Fox, R. H.
"On the Complementary Domains of a Certain Pair of Inequivalent Knots."
Proc. Kon. Nederl. Akad. Wetensch. Ser. A55, 37-40, 1952.Kauffman,
L. Knots
and Physics. Teaneck, NJ: World Scientific, pp. 6-8 and 327, 1991.Rolfsen,
D. Knots
and Links. Wilmington, DE: Publish or Perish Press, pp. 40, 62, and
220, 1976.
as the square
knot. It also has the same group as the square knot
(Rolfsen 1976, p. 62). However, the two knots are distinct, and Fox (1952) showed
that the knot complements of the square and granny
knots are not homeomorphic (Rolfsen 1976, p. 62).
In addition, the square knot is a slice
knot, while the granny knot is not (Rolfsen 1976, p. 220).
