# L8n1

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n1 at Knotilus! L8n1 is $8^2_{16}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v^3-2 u v^2-2 v+1}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $\frac{2}{q^{9/2}}-\frac{2}{q^{7/2}}+\frac{2}{q^{5/2}}-\frac{2}{q^{3/2}}-\frac{2}{q^{11/2}}-\sqrt{q}+\frac{1}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^7 z^{-1} -a^5 z^3-3 a^5 z-2 a^5 z^{-1} +a^3 z^5+4 a^3 z^3+4 a^3 z+2 a^3 z^{-1} -a z^3-3 a z-a z^{-1}$ (db) Kauffman polynomial $3 a^7 z-a^7 z^{-1} +a^6 z^4+2 a^5 z^5-6 a^5 z^3+7 a^5 z-2 a^5 z^{-1} +a^4 z^6-2 a^4 z^4+a^4 z^2-a^4+3 a^3 z^5-10 a^3 z^3+8 a^3 z-2 a^3 z^{-1} +a^2 z^6-3 a^2 z^4+a^2 z^2+a z^5-4 a z^3+4 a z-a z^{-1}$ (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 \
-4-3-2-1012χ
2      11
0       0
-2    21 1
-4   11  0
-6  11   0
-8 11    0
-1011     0
-122      2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-4$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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