# L11n355

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n355 at Knotilus!

 Planar diagram presentation X6172 X10,3,11,4 X18,12,19,11 X7,16,8,17 X15,8,16,9 X20,13,21,14 X22,20,15,19 X12,21,13,22 X14,18,5,17 X2536 X4,9,1,10 Gauss code {1, -10, 2, -11}, {-5, 4, 9, -3, 7, -6, 8, -7}, {10, -1, -4, 5, 11, -2, 3, -8, 6, -9}
A Braid Representative

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(w-1)\left(uv^{2}w-uv^{2}-2uvw+3uv+uw-u-v^{2}w+v^{2}+3vw-2v-w+1\right)}{{\sqrt {u}}vw}}}$ (db) Jones polynomial ${\displaystyle 2q^{-7}-4q^{-6}+9q^{-5}-10q^{-4}+13q^{-3}-q^{2}-12q^{-2}+4q+10q^{-1}-7}$ (db) Signature -2 (db) HOMFLY-PT polynomial ${\displaystyle a^{8}z^{-2}+z^{2}a^{6}-2a^{6}z^{-2}-2a^{6}-2z^{4}a^{4}-2z^{2}a^{4}+a^{4}z^{-2}+a^{4}+z^{6}a^{2}+3z^{4}a^{2}+4z^{2}a^{2}+a^{2}-z^{4}-z^{2}}$ (db) Kauffman polynomial ${\displaystyle 3a^{8}z^{4}-6a^{8}z^{2}-a^{8}z^{-2}+4a^{8}+a^{7}z^{7}+a^{7}z^{5}+a^{7}z^{3}-5a^{7}z+2a^{7}z^{-1}+2a^{6}z^{8}-2a^{6}z^{6}+7a^{6}z^{4}-10a^{6}z^{2}-2a^{6}z^{-2}+6a^{6}+a^{5}z^{9}+4a^{5}z^{7}-7a^{5}z^{5}+7a^{5}z^{3}-5a^{5}z+2a^{5}z^{-1}+6a^{4}z^{8}-7a^{4}z^{6}+2a^{4}z^{4}-a^{4}z^{2}-a^{4}z^{-2}+2a^{4}+a^{3}z^{9}+9a^{3}z^{7}-20a^{3}z^{5}+11a^{3}z^{3}+4a^{2}z^{8}-a^{2}z^{6}-9a^{2}z^{4}+5a^{2}z^{2}-a^{2}+6az^{7}-11az^{5}+z^{5}a^{-1}+4az^{3}-z^{3}a^{-1}+4z^{6}-7z^{4}+2z^{2}}$ (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
5         1-1
3        3 3
1       41 -3
-1      63  3
-3     75   -2
-5    65    1
-7   58     3
-9  45      -1
-11 16       5
-1313        -2
-152         2
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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