# L8a10

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

 Symmetric version Mongolian ornament

 Planar diagram presentation X8192 X10,3,11,4 X12,15,13,16 X14,5,15,6 X4,13,5,14 X16,11,7,12 X2738 X6,9,1,10 Gauss code {1, -7, 2, -5, 4, -8}, {7, -1, 8, -2, 6, -3, 5, -4, 3, -6}
A Braid Representative

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {2t(2)t(1)^{2}-t(1)^{2}+2t(2)^{2}t(1)-3t(2)t(1)+2t(1)-t(2)^{2}+2t(2)}{t(1)t(2)}}}$ (db) Jones polynomial ${\displaystyle -{\frac {1}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}}$ (db) Signature -3 (db) HOMFLY-PT polynomial ${\displaystyle a^{9}z-a^{7}z^{3}+a^{7}z^{-1}-2a^{5}z^{3}-3a^{5}z-a^{5}z^{-1}-a^{3}z^{3}-a^{3}z}$ (db) Kauffman polynomial ${\displaystyle -z^{5}a^{11}+3z^{3}a^{11}-2za^{11}-2z^{6}a^{10}+6z^{4}a^{10}-4z^{2}a^{10}-z^{7}a^{9}+4z^{3}a^{9}-za^{9}-4z^{6}a^{8}+8z^{4}a^{8}-3z^{2}a^{8}-z^{7}a^{7}-2z^{5}a^{7}+6z^{3}a^{7}-4za^{7}+a^{7}z^{-1}-2z^{6}a^{6}+2z^{2}a^{6}-a^{6}-3z^{5}a^{5}+4z^{3}a^{5}-4za^{5}+a^{5}z^{-1}-2z^{4}a^{4}+z^{2}a^{4}-z^{3}a^{3}+za^{3}}$ (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 \
-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
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-4}$ ${\displaystyle i=-2}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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