# L8n6

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n6 at Knotilus! L8n6 is $8^3_{10}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(3)-1) (t(3)+1) (t(1) t(2)+t(3))}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db) Jones polynomial $q^{-2} + q^{-6} + q^{-7} + q^{-9}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^{10} z^{-2} -2 a^8 z^{-2} -2 a^8+a^6 z^{-2} +z^4 a^4+4 z^2 a^4+2 a^4$ (db) Kauffman polynomial $a^{10} z^6-6 a^{10} z^4+10 a^{10} z^2+a^{10} z^{-2} -6 a^{10}+a^9 z^5-6 a^9 z^3+8 a^9 z-2 a^9 z^{-1} +a^8 z^6-7 a^8 z^4+14 a^8 z^2+2 a^8 z^{-2} -9 a^8+a^7 z^5-6 a^7 z^3+8 a^7 z-2 a^7 z^{-1} +a^6 z^{-2} -2 a^6+a^4 z^4-4 a^4 z^2+2 a^4$ (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
-3        11
-5        11
-7     11  0
-9         0
-11   131   1
-13    2    2
-15  1      1
-171        1
-191        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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