# L8n5

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n5 at Knotilus! L8n5 is is $8^3_{9}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)-1) (t(3)-1)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)}}$ (db) Jones polynomial $q^{-7} -2 q^{-6} +3 q^{-5} -2 q^{-4} +4 q^{-3} -2 q^{-2} +2 q^{-1}$ (db) Signature -2 (db) HOMFLY-PT polynomial $a^6 z^2+a^6 z^{-2} +a^6-a^4 z^4-3 a^4 z^2-2 a^4 z^{-2} -4 a^4+2 a^2 z^2+a^2 z^{-2} +3 a^2$ (db) Kauffman polynomial $a^8 z^4-2 a^8 z^2+a^8+2 a^7 z^5-4 a^7 z^3+a^6 z^6-2 a^6 z^2+a^6 z^{-2} -a^6+3 a^5 z^5-5 a^5 z^3+4 a^5 z-2 a^5 z^{-1} +a^4 z^6-a^4 z^4+3 a^4 z^2+2 a^4 z^{-2} -4 a^4+a^3 z^5-a^3 z^3+4 a^3 z-2 a^3 z^{-1} +3 a^2 z^2+a^2 z^{-2} -3 a^2$ (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 \
-6-5-4-3-2-10χ
-1      22
-3     220
-5    2  2
-7    2  2
-9  32   1
-11 12    1
-13 1     -1
-151      1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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