L6a4

From Knot Atlas

Jump to: navigation, search

L6a3

L6a5

1 Link Presentations
2 Polynomial invariants
3 Khovanov Homology
4 Computer Talk
5 Modifying This Page

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L6a4's page at Knotilus.

Visit L6a4's page at the original Knot Atlas.

[edit L6a4 Quick Notes]

The link L6a4 is $6^3_2$ in the Rolfsen table of links.

It is also known as the "Borromean Link" or the "Borromean Rings". A Brunnian link - no two loops are linked directly together, but all three rings are collectively interlinked [9].

Visit Peter Cromwell's page on the Borromean Rings.

[edit L6a4 Further Notes and Views]

Classic-type Borromean rings diagram with color-coded circles	Medieval-style representation of the Borromean rings, used as an emblem of Lorenzo de Medici in San Pancrazio, Florence[1]	A kolam with 3 cycles [2]	A version of the coat of arms of the Borromeo family
The Colombo Mall in Lisboa [3]	The Borromean rings as a symbol of the Christian Trinity (based on a 13th-century French manuscript)	One version of the Germanic "Valknut"	Coat of arms of Hallsberg, Sweden, with padlocks in Borromean configuration
A "Borromean" bathroom tile (the Diane de Poitiers three interlaced crescents emblem) [4]	Rectangles in three dimensions	A Borromean link at the Fields Institute [5]	Basic black-and-white depiction with minimal central overlap
3D depiction	3D depiction which purports to show simple circular toruses interlinked as Borromean rings (something which is actually geometrically impossible).	Asymmetrical depiction	Interlaced rectangles (Miguni, Fukui, Japan).
Borromean rings interlinked with cross as Christian symbol.	A practical application of the Borromean rings (Ballard Locks, Seattle)	Borromean paper clips [6]	A Borromean link by Dylan Thurston [7]
A Borromean rattle by Sassy [8]

[edit] Link Presentations

[edit Notes on L6a4's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,8,9,7 X_4,12,1,11 X_10,5,11,6 X₈₄₅₃ X_2,9,3,10
Gauss code	{1, -6, 5, -3}, {4, -1, 2, -5}, {6, -4, 3, -2}

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(u-1) (v-1) (w-1)}{\sqrt{u} \sqrt{v} \sqrt{w}}$ (db)
Jones polynomial	$- q 3 - q -3 + 3 q 2 + 3 q -2 -2 q -2 q -1 + 4$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- a 2 z 2 - z 2 a -2 + a 2 z -2 + a -2 z -2 + z 4 + 2 z 2 -2 z -2$ (db)
Kauffman polynomial	$a 3 z 3 + z 3 a -3 + 3 a 2 z 4 + 3 z 4 a -2 -4 a 2 z 2 -4 z 2 a -2 + a 2 z -2 + a -2 z -2 + 2 a z 5 + 2 z 5 a -1 - a z 3 - z 3 a -1 -2 a z -1 -2 a -1 z -1 + 6 z 4 -8 z 2 + 2 z -2 + 1$ (db)

[edit] Khovanov Homology

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

0 is the signature of L6a4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:L6a4/KhovanovTable

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -3$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 0$	${\mathbb Z}^{4}$	${\mathbb Z}^{4}$
$r = 1$	${\mathbb Z}^{2}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$