# L11n431

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n431 at Knotilus!

 Planar diagram presentation X8192 X20,10,21,9 X5,15,6,14 X12,14,7,13 X16,8,17,7 X22,18,13,17 X10,4,11,3 X18,11,19,12 X15,1,16,6 X4,20,5,19 X2,21,3,22 Gauss code {1, -11, 7, -10, -3, 9}, {5, -1, 2, -7, 8, -4}, {4, 3, -9, -5, 6, -8, 10, -2, 11, -6}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {(w-1)\left(-u^{2}v^{2}w+u^{2}v^{2}+u^{2}vw-u^{2}v+uv^{2}w-uv^{2}+uvw^{2}-3uvw+uv-uw^{2}+uw-vw^{2}+vw+w^{2}-w\right)}{uvw^{3/2}}}}$ (db) Jones polynomial ${\displaystyle q^{9}-4q^{8}+7q^{7}-9q^{6}+12q^{5}-11q^{4}+11q^{3}-7q^{2}+5q-1}$ (db) Signature 2 (db) HOMFLY-PT polynomial ${\displaystyle z^{2}a^{-8}-2z^{4}a^{-6}-3z^{2}a^{-6}+a^{-6}z^{-2}+z^{6}a^{-4}+3z^{4}a^{-4}+3z^{2}a^{-4}-2a^{-4}z^{-2}-2a^{-4}-z^{4}a^{-2}+a^{-2}z^{-2}+2a^{-2}}$ (db) Kauffman polynomial ${\displaystyle z^{6}a^{-10}-2z^{4}a^{-10}+4z^{7}a^{-9}-11z^{5}a^{-9}+5z^{3}a^{-9}+6z^{8}a^{-8}-19z^{6}a^{-8}+15z^{4}a^{-8}-4z^{2}a^{-8}+3z^{9}a^{-7}-3z^{7}a^{-7}-10z^{5}a^{-7}+8z^{3}a^{-7}+11z^{8}a^{-6}-36z^{6}a^{-6}+37z^{4}a^{-6}-11z^{2}a^{-6}+a^{-6}z^{-2}-2a^{-6}+3z^{9}a^{-5}-5z^{7}a^{-5}+z^{5}a^{-5}+2z^{3}a^{-5}+2za^{-5}-2a^{-5}z^{-1}+5z^{8}a^{-4}-16z^{6}a^{-4}+25z^{4}a^{-4}-10z^{2}a^{-4}+2a^{-4}z^{-2}-3a^{-4}+2z^{7}a^{-3}+2za^{-3}-2a^{-3}z^{-1}+5z^{4}a^{-2}-3z^{2}a^{-2}+a^{-2}z^{-2}-2a^{-2}+z^{3}a^{-1}}$ (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 \
-1012345678χ
19         11
17        3 -3
15       41 3
13      53  -2
11     74   3
9    56    1
7   66     0
5  37      4
3 24       -2
1 4        4
-11         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=1}$ ${\displaystyle i=3}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=8}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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