# L11n433

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n433 at Knotilus!

 Planar diagram presentation X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X20,16,21,15 X5,17,6,16 X11,19,12,18 X17,11,18,10 X19,5,20,4 Gauss code {1, -4, 2, 11, -8, -6}, {4, -1, 5, 10, -9, -3}, {3, -2, 7, 8, -10, 9, -11, -7, 6, -5}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {(uvw+1)\left(uvw^{2}-uw^{2}+u-vw^{2}+v-1\right)}{uvw^{3/2}}}}$ (db) Jones polynomial ${\displaystyle q^{10}-q^{9}+q^{8}-q^{7}+q^{6}+q^{5}+2q^{3}-q^{2}+q}$ (db) Signature 5 (db) HOMFLY-PT polynomial ${\displaystyle -z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-16z^{4}a^{-6}+6z^{4}a^{-8}+10z^{2}a^{-4}-17z^{2}a^{-6}+8z^{2}a^{-8}-z^{2}a^{-10}+6a^{-4}-9a^{-6}+3a^{-8}+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}}$ (db) Kauffman polynomial ${\displaystyle z^{4}a^{-12}-3z^{2}a^{-12}+z^{5}a^{-11}-3z^{3}a^{-11}+z^{6}a^{-10}-5z^{4}a^{-10}+5z^{2}a^{-10}-2a^{-10}+z^{7}a^{-9}-6z^{5}a^{-9}+7z^{3}a^{-9}+2z^{8}a^{-8}-13z^{6}a^{-8}+22z^{4}a^{-8}-11z^{2}a^{-8}-a^{-8}z^{-2}+3a^{-8}+z^{9}a^{-7}-5z^{7}a^{-7}+z^{5}a^{-7}+13z^{3}a^{-7}-9za^{-7}+2a^{-7}z^{-1}+3z^{8}a^{-6}-21z^{6}a^{-6}+44z^{4}a^{-6}-35z^{2}a^{-6}-2a^{-6}z^{-2}+11a^{-6}+z^{9}a^{-5}-6z^{7}a^{-5}+8z^{5}a^{-5}+3z^{3}a^{-5}-9za^{-5}+2a^{-5}z^{-1}+z^{8}a^{-4}-7z^{6}a^{-4}+16z^{4}a^{-4}-16z^{2}a^{-4}-a^{-4}z^{-2}+7a^{-4}}$ (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 \
-2-1012345678χ
21          11
19         110
17       11  0
15      121  0
13     121   0
11    112    2
9   131     1
7  112      2
5 12        1
3           0
11          1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=3}$ ${\displaystyle i=5}$ ${\displaystyle i=7}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=8}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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