Solomon's Seal Knot

Solomon's seal knot is the prime (5,2)-torus knot  with braid word . 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

(1)

(Livingston 1993, p. 127).

The Alexander polynomial , BLM/Ho polynomial , Conway polynomial , HOMFLY polynomial , Jones polynomial , and Kauffman polynomial F  of the Solomon's seal knot are

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

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.

REFERENCES:

Bar-Natan, D. "The Knot ." https://www.math.toronto.edu/~drorbn/KAtlas/Knots/5.1.html.

KnotPlot. "." https://newweb.cecm.sfu.ca/cgi-bin/KnotPlot/KnotServer/kserver?ncomp=1&ncross=5&id=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.