# L11n66

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n66 at Knotilus!

 Planar diagram presentation X6172 X3,10,4,11 X7,14,8,15 X15,22,16,5 X9,17,10,16 X21,9,22,8 X17,21,18,20 X13,18,14,19 X19,12,20,13 X2536 X11,4,12,1 Gauss code {1, -10, -2, 11}, {10, -1, -3, 6, -5, 2, -11, 9, -8, 3, -4, 5, -7, 8, -9, 7, -6, 4}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {uv^{5}-5uv^{4}+8uv^{3}-4uv^{2}+uv+v^{4}-4v^{3}+8v^{2}-5v+1}{{\sqrt {u}}v^{5/2}}}}$ (db) Jones polynomial ${\displaystyle {\frac {12}{q^{9/2}}}-{\frac {13}{q^{7/2}}}+{\frac {12}{q^{5/2}}}+q^{3/2}-{\frac {11}{q^{3/2}}}-{\frac {2}{q^{15/2}}}+{\frac {5}{q^{13/2}}}-{\frac {9}{q^{11/2}}}-4{\sqrt {q}}+{\frac {7}{\sqrt {q}}}}$ (db) Signature -3 (db) HOMFLY-PT polynomial ${\displaystyle a^{9}z^{-1}-a^{7}z^{3}-4a^{7}z-3a^{7}z^{-1}+2a^{5}z^{5}+7a^{5}z^{3}+8a^{5}z+4a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-6a^{3}z^{3}-6a^{3}z-2a^{3}z^{-1}+az^{5}+2az^{3}}$ (db) Kauffman polynomial ${\displaystyle 3a^{9}z^{3}-4a^{9}z+a^{9}z^{-1}+a^{8}z^{6}+4a^{8}z^{4}-4a^{8}z^{2}+a^{8}+5a^{7}z^{7}-13a^{7}z^{5}+25a^{7}z^{3}-16a^{7}z+3a^{7}z^{-1}+6a^{6}z^{8}-16a^{6}z^{6}+23a^{6}z^{4}-12a^{6}z^{2}+3a^{6}+2a^{5}z^{9}+8a^{5}z^{7}-37a^{5}z^{5}+49a^{5}z^{3}-25a^{5}z+4a^{5}z^{-1}+11a^{4}z^{8}-29a^{4}z^{6}+23a^{4}z^{4}-9a^{4}z^{2}+2a^{4}+2a^{3}z^{9}+7a^{3}z^{7}-35a^{3}z^{5}+35a^{3}z^{3}-15a^{3}z+2a^{3}z^{-1}+5a^{2}z^{8}-11a^{2}z^{6}+2a^{2}z^{4}+a^{2}+4az^{7}-11az^{5}+8az^{3}-2az+z^{6}-2z^{4}+z^{2}}$ (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 \
-6-5-4-3-2-10123χ
4         1-1
2        3 3
0       41 -3
-2      73  4
-4     65   -1
-6    76    1
-8   56     1
-10  47      -3
-12 26       4
-14 3        -3
-162         2
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-4}$ ${\displaystyle i=-2}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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