# L8a10

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a10 at Knotilus! L8a10 is $8^2_{5}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 t(2) t(1)^2-t(1)^2+2 t(2)^2 t(1)-3 t(2) t(1)+2 t(1)-t(2)^2+2 t(2)}{t(1) t(2)}$ (db) Jones polynomial $-\frac{1}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 z-a^7 z^3+a^7 z^{-1} -2 a^5 z^3-3 a^5 z-a^5 z^{-1} -a^3 z^3-a^3 z$ (db) Kauffman polynomial $-z^5 a^{11}+3 z^3 a^{11}-2 z a^{11}-2 z^6 a^{10}+6 z^4 a^{10}-4 z^2 a^{10}-z^7 a^9+4 z^3 a^9-z a^9-4 z^6 a^8+8 z^4 a^8-3 z^2 a^8-z^7 a^7-2 z^5 a^7+6 z^3 a^7-4 z a^7+a^7 z^{-1} -2 z^6 a^6+2 z^2 a^6-a^6-3 z^5 a^5+4 z^3 a^5-4 z a^5+a^5 z^{-1} -2 z^4 a^4+z^2 a^4-z^3 a^3+z a^3$ (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 \
-8-7-6-5-4-3-2-10χ
-2        11
-4       21-1
-6      2  2
-8     22  0
-10    32   1
-12   12    1
-14  23     -1
-16 12      1
-18 1       -1
-201        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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