6₂ knot

6₂ knot
Arf invariant	1
Braid length	6
Braid no.	3
Bridge no.	2
Crosscap no.	2
Crossing no.	6
Genus	2
Hyperbolic volume	4.40083
Stick no.	8
Unknotting no.	1
Conway notation	[312]
A-B notation	62
Dowker notation	4, 8, 10, 12, 2, 6
Last /Next	61 / 63
Other
alternating, hyperbolic, fibered, prime, reversible, twist

In knot theory, the 6₂ knot is one of three prime knots with crossing number six, the others being the stevedore knot and the 6₃ knot. This knot is sometimes referred to as the Miller Institute knot,^[1] because it appears in the logo^[2] of the Miller Institute for Basic Research in Science at the University of California, Berkeley.

The 6₂ knot is invertible but not amphichiral. Its Alexander polynomial is

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

its Conway polynomial is

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

and its Jones polynomial is

$V(q) = q - 1 + 2q^{-1} - 2q^{-2} + 2q^{-3} - 2q^{-4} + q^{-5}. \,$^[3]

The 6₂ knot is a hyperbolic knot, with its complement having a volume of approximately 4.40083.

References

1. Weisstein, Eric W., "Miller Institute Knot", MathWorld.
2. Miller Institute - Home Page
3. "6_2", The Knot Atlas.