# L8a12

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a12 at Knotilus! L8a12 is $8^2_{2}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(2)^2 t(1)^3-t(2)^3 t(1)^2+t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+t(2) t(1)-t(1)-t(2)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{2}{q^{17/2}}+\frac{1}{q^{19/2}}-\frac{1}{q^{21/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^9 z^3+3 a^9 z+a^9 z^{-1} -a^7 z^5-4 a^7 z^3-4 a^7 z-a^7 z^{-1} -a^5 z^5-4 a^5 z^3-3 a^5 z$ (db) Kauffman polynomial $-z^3 a^{13}+2 z a^{13}-z^4 a^{12}+z^2 a^{12}-z^5 a^{11}+z^3 a^{11}-z a^{11}-z^6 a^{10}+2 z^4 a^{10}-3 z^2 a^{10}-z^7 a^9+4 z^5 a^9-8 z^3 a^9+5 z a^9-a^9 z^{-1} -2 z^6 a^8+6 z^4 a^8-5 z^2 a^8+a^8-z^7 a^7+4 z^5 a^7-6 z^3 a^7+5 z a^7-a^7 z^{-1} -z^6 a^6+3 z^4 a^6-z^2 a^6-z^5 a^5+4 z^3 a^5-3 z a^5$ (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 \
-8-7-6-5-4-3-2-10χ
-4        11
-6       110
-8      1  1
-10     11  0
-12    21   1
-14   11    0
-16  12     -1
-18  1      1
-2011       0
-221        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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