63 knot
# 63 knot

In knot theory, the $6_3$ knot is one of three prime knots with crossing number six, the others being the stevedore knot and the $6_2$ knot. It is alternating, hyperbolic, and fully amphichiral. It can be written as the braid word

    $\sigma _{1}^{-1}\sigma _{2}^{2}\sigma _{1}^{-2}\sigma _{2}.\,$

## Symmetry

Like the figure-eight knot, the $6_3$ knot is fully amphichiral. This means that the $6_3$ knot is amphichiral, 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.

## Invariants

The Alexander polynomial of the $6_3$ knot is

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

Conway polynomial is

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

Jones polynomial is

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

and the Kauffman polynomial is

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

The $6_3$ knot is a hyperbolic knot, with its complement having a volume of approximately 5.69302.

