# L8n8

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n8 at Knotilus! L8n8 is $8^4_{3}$ in the Rolfsen table of links.  Detail from an 18th century royal decree, Vietnam.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(2)-t(3)) (t(1)-t(4))}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}}$ (db) Jones polynomial $-q^{7/2}-q^{3/2}-2 \sqrt{q}-\frac{2}{\sqrt{q}}-\frac{1}{q^{3/2}}-\frac{1}{q^{7/2}}$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^3 z^{-3} - a^{-3} z^{-3} +a^3 z+2 a^3 z^{-1} -z a^{-3} -2 a^{-3} z^{-1} -a z^3-3 a z^{-3} +z^3 a^{-1} +3 a^{-1} z^{-3} -5 a z-6 a z^{-1} +5 z a^{-1} +6 a^{-1} z^{-1}$ (db) Kauffman polynomial $-a^3 z^5-a z^5-z^5 a^{-1} -z^5 a^{-3} -a^2 z^4-z^4 a^{-2} -2 z^4+5 a^3 z^3+7 a z^3+7 z^3 a^{-1} +5 z^3 a^{-3} +6 a^2 z^2+6 z^2 a^{-2} +12 z^2-6 a^3 z-14 a z-14 z a^{-1} -6 z a^{-3} -8 a^2-8 a^{-2} -15+4 a^3 z^{-1} +9 a z^{-1} +9 a^{-1} z^{-1} +4 a^{-3} z^{-1} +3 a^2 z^{-2} +3 a^{-2} z^{-2} +6 z^{-2} -a^3 z^{-3} -3 a z^{-3} -3 a^{-1} z^{-3} - a^{-3} z^{-3}$ (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 \
-4-3-2-101234χ
8        11
6        11
4      1  1
2    3    3
0   161   4
-2    3    3
-4  1      1
-61        1
-81        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{6}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ $r=4$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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