Solomon's Seal Knot


Solomon's seal knot is the prime (5,2)-torus knot 5_1 with braid word sigma_1^5. It is also known as the cinquefoil knot (a name derived from certain herbs and shrubs of the rose family which have five-lobed leaves and five-petaled flowers) or the double overhand knot. It has Arf invariant 1 and is not amphichiral, although it is invertible.

The knot group of Solomon's seal knot is


(Livingston 1993, p. 127).

The Alexander polynomial Delta(x), BLM/Ho polynomial Q(x), Conway polynomial del (x), HOMFLY polynomial P(l,m), Jones polynomial V(t), and Kauffman polynomial F F(a,z) of the Solomon's seal knot are

del (x)=x^4+3x^2+1

Surprisingly, the knot 10-132 shares the same Alexander polynomial and Jones polynomial with the Solomon's seal knot. However, no knots on 10 or fewer crossings share the same BLM/Ho polynomial with it.

See also

Figure Eight Knot, Knot, Prime Knot, Trefoil Knot, Torus Knot

Explore with Wolfram|Alpha


Bar-Natan, D. "The Knot 5_1." "5_1.", C. Knot Theory. Washington, DC: Math. Assoc. Amer., 1993.Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 53, 1976.

Cite this as:

Weisstein, Eric W. "Solomon's Seal Knot." From MathWorld--A Wolfram Web Resource.

Subject classifications