6₃ knot

Like the figure-eight knot, the 6₃ knot is fully amphichiral. This means that the 6₃ knot is amphichiral,[2] meaning that it is indistinguishable from its own mirror image. In addition, it is also invertible, meaning that orienting the curve in either direction yields the same oriented knot.

The Alexander polynomial of the 6₃ knot is

${\displaystyle \Delta (t)=t^{2}-3t+5-3t^{-1}+t^{-2},\,}$

Conway polynomial is

${\displaystyle \nabla (z)=z^{4}+z^{2}+1,\,}$

Jones polynomial is

${\displaystyle V(q)=-q^{3}+2q^{2}-2q+3-2q^{-1}+2q^{-2}-q^{-3},\,}$

and the Kauffman polynomial is

${\displaystyle L(a,z)=az^{5}+z^{5}a^{-1}+2a^{2}z^{4}+2z^{4}a^{-2}+4z^{4}+a^{3}z^{3}+az^{3}+z^{3}a^{-1}+z^{3}a^{-3}-3a^{2}z^{2}-3z^{2}a^{-2}-6z^{2}-a^{3}z-2az-2za^{-1}-za^{}-3+a^{2}+a^{-2}+3.\,}$ [3]

The 6₃ knot is a hyperbolic knot, with its complement having a volume of approximately 5.69302.