6₃ knot

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]

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

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

1. https://www.wolframalpha.com/input/?i=6_3+knot
2. Weisstein, Eric W. "Amphichiral Knot". MathWorld. Accessed: May 12, 2014.
3. "6_3", The Knot Atlas.