6₃ knot

63 knot
Arf invariant	1
Braid length	6
Braid no.	3
Bridge no.	2
Crosscap no.	3
Crossing no.	6
Genus	2
Hyperbolic volume	5.69302
Stick no.	8
Unknotting no.	1
Conway notation	[2112]
A–B notation	63
Dowker notation	4, 8, 10, 2, 12, 6
Last /Next	62 / 71
Other
alternating, hyperbolic, fibered, prime, fully amphichiral

In knot theory, the 6₃ knot is one of three prime knots with crossing number six, the others being the stevedore knot and the 6₂ knot. It is alternating, hyperbolic, and fully amphichiral. It can be written as the braid word

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

Symmetry[]

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.

Invariants[]

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.

References[]