# L8n5

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n5 at Knotilus! L8n5 is is ${\displaystyle 8_{9}^{3}}$ in the Rolfsen table of links.

 Planar diagram presentation X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X9,16,10,11 X11,10,12,5 X4,15,1,16 Gauss code {1, -4, 3, -8}, {-2, -1, 5, -3, -6, 7}, {-7, 2, 4, -5, 8, 6}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {(t(1)-1)(t(2)-1)(t(3)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}}}}$ (db) Jones polynomial ${\displaystyle q^{-7}-2q^{-6}+3q^{-5}-2q^{-4}+4q^{-3}-2q^{-2}+2q^{-1}}$ (db) Signature -2 (db) HOMFLY-PT polynomial ${\displaystyle a^{6}z^{2}+a^{6}z^{-2}+a^{6}-a^{4}z^{4}-3a^{4}z^{2}-2a^{4}z^{-2}-4a^{4}+2a^{2}z^{2}+a^{2}z^{-2}+3a^{2}}$ (db) Kauffman polynomial ${\displaystyle a^{8}z^{4}-2a^{8}z^{2}+a^{8}+2a^{7}z^{5}-4a^{7}z^{3}+a^{6}z^{6}-2a^{6}z^{2}+a^{6}z^{-2}-a^{6}+3a^{5}z^{5}-5a^{5}z^{3}+4a^{5}z-2a^{5}z^{-1}+a^{4}z^{6}-a^{4}z^{4}+3a^{4}z^{2}+2a^{4}z^{-2}-4a^{4}+a^{3}z^{5}-a^{3}z^{3}+4a^{3}z-2a^{3}z^{-1}+3a^{2}z^{2}+a^{2}z^{-2}-3a^{2}}$ (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 \
-6-5-4-3-2-10χ
-1      22
-3     220
-5    2  2
-7    2  2
-9  32   1
-11 12    1
-13 1     -1
-151      1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$

### Computer Talk

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