The square knot has the same Alexander polynomial as the granny
knot. It also has the same group as the granny knot
(Rolfsen 1976, p. 62). However, the two knots are distinct. 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.Livingston,
C. Knot
Theory. Washington, DC: Math. Assoc. Amer., pp. 38-39, 1993.Owen,
P. Knots.
Philadelphia, PA: Courage, p. 50, 1993.Rolfsen, D. Knots
and Links. Wilmington, DE: Publish or Perish Press, pp. 40, 61, and
220, 1976.
as the granny
knot. It also has the same group as the granny knot
(Rolfsen 1976, p. 62). However, the two knots are distinct. 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).
