62 knot

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title: "62 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/62_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	62 knot
image	Blue 6 2 Knot.png
arf invariant	1
braid length	6
braid number	3
bridge number	2
crosscap number	2
crossing number	6
genus	2
hyperbolic volume	4.40083
stick number	8
unknotting number	1
conway_notation	[312]
ab_notation	62
dowker notation	4, 8, 10, 12, 2, 6
last crossing	6
last order	1
next crossing	6
next order	3
alternating	alternating
class	hyperbolic
fibered	fibered
prime	prime
symmetry	reversible
::

| name= 62 knot | practical name= | image= Blue 6 2 Knot.png | caption= | arf invariant= 1 | braid length= 6 | braid number= 3 | bridge number= 2 | crosscap number= 2 | crossing number= 6 | genus= 2 | hyperbolic volume= 4.40083 | linking number= | stick number= 8 | unknotting number= 1 | conway_notation= [312] | ab_notation= 62 | dowker notation= 4, 8, 10, 12, 2, 6 | thistlethwaite= | last crossing= 6 | last order= 1 | next crossing= 6 | next order= 3 | alternating= alternating | class= hyperbolic | fibered= fibered | prime= prime | slice= | symmetry= reversible | tricolorable= | 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, because it appears in the logo 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 + 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}. ,

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

Surface

File:Superfície - bordo Nó 6,2.jpg|Surface of knot 6.2

Example

Ways to assemble of knot 6.2 File:6₂ knot.webm|Example 1 File:6₂ knot (2).webm|Example 2

If a bowline is tied and the two free ends of the rope are brought together in the simplest way, the knot obtained is the 62 knot. The sequence of necessary moves are depicted here: Image:Bowline to 6 2 knot.gif|From a bowline (ends connected) to the 6₂ knot.

References

References

1. "Miller Institute Knot".
2. [http://millerinstitute.berkeley.edu Miller Institute - Home Page]
3. {{Knot Atlas. 6_2

::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. ::