Notice. The Knot Atlas is now recovering from a major crash. Hopefully all functionality will return slowly over the next few days. --Drorbn (talk) 21:23, 4 July 2013 (EDT)

L6a4

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L6a3.gif

L6a3

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L6a5

Contents

L6a4.gif
(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.

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.


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]


Link Presentations

[edit Notes on L6a4's Link Presentations]

Planar diagram presentation X6172 X12,8,9,7 X4,12,1,11 X10,5,11,6 X8453 X2,9,3,10
Gauss code {1, -6, 5, -3}, {4, -1, 2, -5}, {6, -4, 3, -2}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L6a4 ML.gif

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)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=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}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

Back to the top.

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L6a3

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L6a5