# 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 . 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  A version of the coat of arms of the Borromeo family  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)  A Borromean link at the Fields Institute  Basic black-and-white depiction with minimal central overlap  3D depiction which purports to show simple circular toruses interlinked as Borromean rings (something which is actually geometrically impossible).  Interlaced rectangles (Miguni, Fukui, Japan).  Borromean rings interlinked with cross as Christian symbol.  A practical application of the Borromean rings (Ballard Locks, Seattle)

### 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.