Granny Knot

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).

The knot group of the granny knot is

which is isomorphic to that of the square knot (Rolfsen 1976, p. 62)

REFERENCES:

Cipra, B. A. "To Have and Have Knot: When Are Knots Alike?" Science 241, 1291-1292, 1988.

Fox, R. H. "On the Complementary Domains of a Certain Pair of Inequivalent Knots." Proc. Kon. Nederl. Akad. Wetensch. Ser. A 55, 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.