# L6a4

## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L6a4 at Knotilus! 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]

 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}

### 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 $r$).
 \ r \ j \
-3-2-10123χ
7      1-1
5     2 2
3     1 1
1   42  2
-1  24   2
-3 1     1
-5 2     2
-71      -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{4}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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