# L8n8

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n8 at Knotilus! L8n8 is ${\displaystyle 8_{3}^{4}}$ in the Rolfsen table of links.
 Detail from an 18th century royal decree, Vietnam.

 Planar diagram presentation X6172 X2536 X16,11,13,12 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X12,15,9,16 Gauss code {1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 3, -8}, {-7, 6, 8, -3}
A Braid Representative

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {(t(2)-t(3))(t(1)-t(4))}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}{\sqrt {t(4)}}}}}$ (db) Jones polynomial ${\displaystyle -q^{7/2}-q^{3/2}-2{\sqrt {q}}-{\frac {2}{\sqrt {q}}}-{\frac {1}{q^{3/2}}}-{\frac {1}{q^{7/2}}}}$ (db) Signature 0 (db) HOMFLY-PT polynomial ${\displaystyle a^{3}z^{-3}-a^{-3}z^{-3}+a^{3}z+2a^{3}z^{-1}-za^{-3}-2a^{-3}z^{-1}-az^{3}-3az^{-3}+z^{3}a^{-1}+3a^{-1}z^{-3}-5az-6az^{-1}+5za^{-1}+6a^{-1}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle -a^{3}z^{5}-az^{5}-z^{5}a^{-1}-z^{5}a^{-3}-a^{2}z^{4}-z^{4}a^{-2}-2z^{4}+5a^{3}z^{3}+7az^{3}+7z^{3}a^{-1}+5z^{3}a^{-3}+6a^{2}z^{2}+6z^{2}a^{-2}+12z^{2}-6a^{3}z-14az-14za^{-1}-6za^{-3}-8a^{2}-8a^{-2}-15+4a^{3}z^{-1}+9az^{-1}+9a^{-1}z^{-1}+4a^{-3}z^{-1}+3a^{2}z^{-2}+3a^{-2}z^{-2}+6z^{-2}-a^{3}z^{-3}-3az^{-3}-3a^{-1}z^{-3}-a^{-3}z^{-3}}$ (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 \
-4-3-2-101234χ
8        11
6        11
4      1  1
2    3    3
0   161   4
-2    3    3
-4  1      1
-61        1
-81        1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle i=2}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=3}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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