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 | 6_{3} |
Dowker notation | 4, 8, 10, 2, 12, 6 |
Last /Next | 6_{2} / 7_{1} |
Other | |
alternating, hyperbolic, fibered, prime, fully amphichiral |
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
- ^{[1]}
Symmetry
Like the figure-eight knot, the 6_{3} knot is fully amphichiral. This means that the 6_{3} 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 6_{3} knot is
and the Kauffman polynomial is
- ^{[3]}
The 6_{3} knot is a hyperbolic knot, with its complement having a volume of approximately 5.69302.
References
- ^ https://www.wolframalpha.com/input/?i=6_3+knot
- ^ Weisstein, Eric W. "Amphichiral Knot". MathWorld. Accessed: May 12, 2014.
- ^ "6_3", The Knot Atlas.