63 knot

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title: "63 knot" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["double-torus-knots-and-links"] description: "Mathematical knot with crossing number 6" topic_path: "general/double-torus-knots-and-links" source: "https://en.wikipedia.org/wiki/63_knot" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Mathematical knot with crossing number 6 ::

::data[format=table title="Infobox knot theory"]

Field	Value
name	63 knot
image	Blue 6 3 Knot.png
arf invariant	1
braid length	6
braid number	3
bridge number	2
crosscap number	3
crossing number	6
genus	2
hyperbolic volume	5.69302
stick number	8
unknotting number	1
conway_notation	[2112]
ab_notation	63
dowker notation	4, 8, 10, 2, 12, 6
last crossing	6
last order	2
next crossing	7
next order	1
alternating	alternating
class	hyperbolic
fibered	fibered
prime	prime
symmetry	fully amphichiral
::

| name= 63 knot | practical name= | image= Blue 6 3 Knot.png | caption= | arf invariant= 1 | braid length= 6 | braid number= 3 | bridge number= 2 | crosscap number= 3 | crossing number= 6 | genus= 2 | hyperbolic volume= 5.69302 | linking number= | stick number= 8 | unknotting number= 1 | conway_notation= [2112] | ab_notation= 63 | dowker notation= 4, 8, 10, 2, 12, 6 | thistlethwaite= | last crossing= 6 | last order= 2 | next crossing= 7 | next order= 1 | alternating= alternating | class= hyperbolic | fibered= fibered | prime= prime | slice= | symmetry= fully amphichiral | tricolorable= | twist= 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 :\sigma_1^{-1}\sigma_2^2\sigma_1^{-2}\sigma_2. ,

Symmetry

Like the figure-eight knot, the 63 knot is fully amphichiral. This means that the 63 knot is amphichiral, 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 :\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. ,

The 63 knot is a hyperbolic knot, with its complement having a volume of approximately 5.69302.

References

References

1. "6_3 knot - Wolfram|Alpha".
2. "Amphichiral Knot".
3. {{Knot Atlas. 6_3

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::