These are the same as the Alexander, Conway, and Jones polynomials of the knot 10132. However, the Kauffman polynomial can be used to distinguish between these two knots.
Historyedit
The name “cinquefoil” comes from the five-petaled flowers of plants in the genus Potentilla.
cinquefoil, knot, knot, theory, cinquefoil, knot, also, known, solomon, seal, knot, pentafoil, knot, knots, with, crossing, number, five, other, being, three, twist, knot, listed, knot, alexander, briggs, notation, also, described, torus, knot, cinquefoil, clo. In knot theory the cinquefoil knot also known as Solomon s seal knot or the pentafoil knot is one of two knots with crossing number five the other being the three twist knot It is listed as the 51 knot in the Alexander Briggs notation and can also be described as the 5 2 torus knot The cinquefoil is the closed version of the double overhand knot CinquefoilCommon nameDouble overhand knotArf invariant1Braid length5Braid no 2Bridge no 2Crosscap no 1Crossing no 5Genus2Hyperbolic volume0Stick no 8Unknotting no 2Conway notation 5 A B notation51Dowker notation6 8 10 2 4Last Next41 52Otheralternating torus fibered prime reversible Contents 1 Properties 2 History 3 See also 4 References 5 Further readingProperties editThe cinquefoil is a prime knot Its writhe is 5 and it is invertible but not amphichiral 1 Its Alexander polynomial is D t t 2 t 1 t 1 t 2 displaystyle Delta t t 2 t 1 t 1 t 2 nbsp its Conway polynomial is z z 4 3 z 2 1 displaystyle nabla z z 4 3z 2 1 nbsp and its Jones polynomial is V q q 2 q 4 q 5 q 6 q 7 displaystyle V q q 2 q 4 q 5 q 6 q 7 nbsp These are the same as the Alexander Conway and Jones polynomials of the knot 10132 However the Kauffman polynomial can be used to distinguish between these two knots History editThe name cinquefoil comes from the five petaled flowers of plants in the genus Potentilla source source source source source source source source Assembling of Cinquefoil knot nbsp Edible cinquefoil knot See also editPentagram Trefoil knot 7 knot Skein relationReferences edit Weisstein Eric W Solomon s Seal Knot MathWorld Further reading editA Pentafoil Knot at the Wayback Machine archived June 4 2004 Retrieved from https en wikipedia org w index php title Cinquefoil knot amp oldid 1037300574, wikipedia, wiki, book, books, library,