Unital

In knot theory, the three-twist knot is the twist knot with three-half twists. It is listed as the 5₂ knot in the Alexander-Briggs notation, and is one of two knots with crossing number five, the other being the cinquefoil knot.

The three-twist knot is a prime knot, and it is invertible but not amphichiral. Its Alexander polynomial is

$\mathrm{\Delta }(t)=2t-3+2t^{-1},$

its Conway polynomialTemplate:Dn is

$\mathrm{\nabla }(z)=2z^2+1,$

and its Jones polynomial is

$V(q)=q^{-1}-q^{-2}+2q^{-3}-q^{-4}+q^{-5}-q^{-6}.$

Because the Alexander polynomial is not monic, the three-twist knot is not fibered.

The three-twist knot is a hyperbolic knot, with its complement having a volume of approximately 2.82812.

References

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