6₃ knot

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6₃ knot
Blue 6 3 Knot.png
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 6271
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 form the braid word

\sigma_1^{-1}\sigma_2^2\sigma_1^{-2}\sigma_2. \, [1]

Symmetry[edit]

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[edit]

The Alexander polynomial of the 63 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^5a^{-1} + 2a^2z^4 + 2z^4a^{-2} + 4z^4 + a^3z^3 + az^3 + z^3a^{-1} + z^3a^{-3} - 3a^2z^2 - 3z^2a^{-2} - 6z^2 - a^3z - 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[edit]