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
| <x,y|xyxyxy^(-1)x^(-1)y^(-1)x^(-1)y^(-1)> |
(1)
|
(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
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 KnotExplore with Wolfram|Alpha
More things to try:
References
Bar-Natan, D. "The Knot 5_1." https://katlas.org/wiki/5_1.Livingston, C. Knot Theory. Washington, DC: Math. Assoc. Amer., 1993.Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 53, 1976.Referenced on Wolfram|Alpha
Solomon's Seal KnotCite this as:
Weisstein, Eric W. "Solomon's Seal Knot." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SolomonsSealKnot.html