# L8n6

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

 Planar diagram presentation X6172 X10,3,11,4 X11,16,12,13 X7,14,8,15 X13,8,14,9 X15,12,16,5 X2536 X4,9,1,10 Gauss code {1, -7, 2, -8}, {-5, 4, -6, 3}, {7, -1, -4, 5, 8, -2, -3, 6}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(t(3)-1)(t(3)+1)(t(1)t(2)+t(3))}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}}$ (db) Jones polynomial ${\displaystyle q^{-2}+q^{-6}+q^{-7}+q^{-9}}$ (db) Signature -3 (db) HOMFLY-PT polynomial ${\displaystyle a^{10}z^{-2}-2a^{8}z^{-2}-2a^{8}+a^{6}z^{-2}+z^{4}a^{4}+4z^{2}a^{4}+2a^{4}}$ (db) Kauffman polynomial ${\displaystyle a^{10}z^{6}-6a^{10}z^{4}+10a^{10}z^{2}+a^{10}z^{-2}-6a^{10}+a^{9}z^{5}-6a^{9}z^{3}+8a^{9}z-2a^{9}z^{-1}+a^{8}z^{6}-7a^{8}z^{4}+14a^{8}z^{2}+2a^{8}z^{-2}-9a^{8}+a^{7}z^{5}-6a^{7}z^{3}+8a^{7}z-2a^{7}z^{-1}+a^{6}z^{-2}-2a^{6}+a^{4}z^{4}-4a^{4}z^{2}+2a^{4}}$ (db)

### Khovanov Homology

The coefficients of the monomials ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$).
 \ 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
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-5}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-7}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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