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Solomon's Seal Knot

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SolomonsSealKnot
SolomonsSealKnot3D

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

Delta(x)=x^2-x+1-x^(-1)+x^(-2)
(2)
Q(x)=2x^4+2x^3-6x^2-2x+5
(3)
del (x)=x^4+3x^2+1
(4)
P(l,m)=m^4l^4+m^2(-l^6-4l^4)+(3l^4+2l^6)
(5)
V(t)=t^2+t^4-t^5+t^6-t^7
(6)
F(a,z)=2a^6+3a^4+(a^6+a^4)z^4+(a^7+a^5)z^3+(a^8-3a^6-4a^4)z^2+(a^9-a^7-2a^5)z.
(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.

See also

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

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References

Bar-Natan, D. "The Knot 5_1." http://www.math.toronto.edu/~drorbn/KAtlas/Knots/5.1.html.KnotPlot. "5_1." http://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.

Cite this as:

Weisstein, Eric W. "Solomon's Seal Knot." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SolomonsSealKnot.html

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