# L10a140

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a140 at Knotilus! Brunnian link. Presumably the simplest Brunnian link other than the Borromean rings. The second in an infinite series of Brunnian links -- if the blue and yellow loops in the illustration below have only one twist along each side, the result is the Borromean rings; if the blue and yellow loops have three twists along each side, the result is this L10a140 link; if the blue and yellow loops have five twists along each side, the result is a three-loop link with 14 overall crossings, etc.  In a visual form which makes it evident that it is a Brunnian link.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) (t(3)-1) (t(2) t(3)+1)^2}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}}$ (db) Jones polynomial $-q^5+3 q^4-5 q^3+8 q^2-9 q+12-9 q^{-1} +8 q^{-2} -5 q^{-3} +3 q^{-4} - q^{-5}$ (db) Signature 0 (db) HOMFLY-PT polynomial $-a^2 z^6-z^6 a^{-2} -4 a^2 z^4-4 z^4 a^{-2} -4 a^2 z^2-4 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +z^8+6 z^6+12 z^4+8 z^2-2 z^{-2}$ (db) Kauffman polynomial $2 a z^9+2 z^9 a^{-1} +4 a^2 z^8+4 z^8 a^{-2} +8 z^8+4 a^3 z^7-2 a z^7-2 z^7 a^{-1} +4 z^7 a^{-3} +3 a^4 z^6-11 a^2 z^6-11 z^6 a^{-2} +3 z^6 a^{-4} -28 z^6+a^5 z^5-9 a^3 z^5-2 a z^5-2 z^5 a^{-1} -9 z^5 a^{-3} +z^5 a^{-5} -7 a^4 z^4+14 a^2 z^4+14 z^4 a^{-2} -7 z^4 a^{-4} +42 z^4-2 a^5 z^3+4 a^3 z^3+6 a z^3+6 z^3 a^{-1} +4 z^3 a^{-3} -2 z^3 a^{-5} +2 a^4 z^2-8 a^2 z^2-8 z^2 a^{-2} +2 z^2 a^{-4} -20 z^2+1-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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 \
-5-4-3-2-1012345χ
11          1-1
9         2 2
7        31 -2
5       52  3
3      43   -1
1     85    3
-1    58     3
-3   34      -1
-5  25       3
-7 13        -2
-9 2         2
-111          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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