# L11a384

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a384 at Knotilus!

 Planar diagram presentation X12,1,13,2 X14,8,15,7 X16,4,17,3 X20,6,21,5 X18,10,19,9 X10,18,1,17 X4,20,5,19 X22,14,11,13 X6,16,7,15 X2,11,3,12 X8,22,9,21 Gauss code {1, -10, 3, -7, 4, -9, 2, -11, 5, -6}, {10, -1, 8, -2, 9, -3, 6, -5, 7, -4, 11, -8}
A Braid Representative

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {u^{4}v^{2}-u^{4}v+3u^{3}v^{3}-8u^{3}v^{2}+5u^{3}v-u^{3}+u^{2}v^{4}-8u^{2}v^{3}+13u^{2}v^{2}-8u^{2}v+u^{2}-uv^{4}+5uv^{3}-8uv^{2}+3uv-v^{3}+v^{2}}{u^{2}v^{2}}}}$ (db) Jones polynomial ${\displaystyle 22q^{9/2}-19q^{7/2}+14q^{5/2}-8q^{3/2}+q^{21/2}-4q^{19/2}+9q^{17/2}-15q^{15/2}+19q^{13/2}-23q^{11/2}+3{\sqrt {q}}-{\frac {1}{\sqrt {q}}}}$ (db) Signature 3 (db) HOMFLY-PT polynomial ${\displaystyle z^{3}a^{-9}-z^{5}a^{-7}+z^{3}a^{-7}+2za^{-7}-a^{-7}z^{-1}-3z^{5}a^{-5}-6z^{3}a^{-5}-3za^{-5}+a^{-5}z^{-1}-z^{5}a^{-3}+z^{3}a^{-3}+3za^{-3}+z^{3}a^{-1}+za^{-1}}$ (db) Kauffman polynomial ${\displaystyle -2z^{10}a^{-6}-2z^{10}a^{-8}-6z^{9}a^{-5}-12z^{9}a^{-7}-6z^{9}a^{-9}-8z^{8}a^{-4}-13z^{8}a^{-6}-12z^{8}a^{-8}-7z^{8}a^{-10}-6z^{7}a^{-3}+2z^{7}a^{-5}+18z^{7}a^{-7}+6z^{7}a^{-9}-4z^{7}a^{-11}-3z^{6}a^{-2}+13z^{6}a^{-4}+33z^{6}a^{-6}+33z^{6}a^{-8}+15z^{6}a^{-10}-z^{6}a^{-12}-z^{5}a^{-1}+9z^{5}a^{-3}+8z^{5}a^{-5}-4z^{5}a^{-7}+7z^{5}a^{-9}+9z^{5}a^{-11}+4z^{4}a^{-2}-13z^{4}a^{-4}-28z^{4}a^{-6}-21z^{4}a^{-8}-8z^{4}a^{-10}+2z^{4}a^{-12}+2z^{3}a^{-1}-7z^{3}a^{-3}-8z^{3}a^{-5}+2z^{3}a^{-7}-5z^{3}a^{-9}-6z^{3}a^{-11}-z^{2}a^{-2}+6z^{2}a^{-4}+10z^{2}a^{-6}+5z^{2}a^{-8}+z^{2}a^{-10}-z^{2}a^{-12}-za^{-1}+3za^{-3}-za^{-5}-5za^{-7}+za^{-9}+za^{-11}-a^{-6}+a^{-5}z^{-1}+a^{-7}z^{-1}}$ (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 \
-2-10123456789χ
22           1-1
20          3 3
18         61 -5
16        93  6
14       117   -4
12      128    4
10     1011     1
8    912      -3
6   510       5
4  39        -6
2 16         5
0 2          -2
-21           1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=2}$ ${\displaystyle i=4}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}}$ ${\displaystyle {\mathbb {Z} }^{9}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}}$ ${\displaystyle {\mathbb {Z} }^{12}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}}$ ${\displaystyle {\mathbb {Z} }^{11}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{9}}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=8}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=9}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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