# L11n1

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n1 at Knotilus!

 Link L11n1. A graph, L11n1. A part of a knot and a part of a graph.

 Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X11,20,12,21 X17,5,18,22 X21,19,22,18 X19,12,20,13 X9,16,10,17 X13,2,14,3 Gauss code {1, 11, -5, -3}, {-4, -1, 2, 5, -10, 4, -6, 9, -11, -2, 3, 10, -7, 8, -9, 6, -8, 7}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {t(1)t(2)^{5}-4t(1)t(2)^{4}+2t(1)t(2)^{3}+2t(2)^{2}-4t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}}$ (db) Jones polynomial ${\displaystyle -{\frac {1}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {3}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {3}{q^{15/2}}}-{\frac {2}{q^{17/2}}}+{\frac {1}{q^{19/2}}}}$ (db) Signature -5 (db) HOMFLY-PT polynomial ${\displaystyle -a^{11}z^{-1}+a^{9}z^{3}+5a^{9}z+3a^{9}z^{-1}-2a^{7}z^{5}-9a^{7}z^{3}-9a^{7}z-3a^{7}z^{-1}+a^{5}z^{7}+5a^{5}z^{5}+7a^{5}z^{3}+5a^{5}z+2a^{5}z^{-1}-a^{3}z^{5}-4a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle -z^{2}a^{12}+a^{12}-2z^{3}a^{11}+2za^{11}-a^{11}z^{-1}-z^{6}a^{10}+3z^{4}a^{10}-5z^{2}a^{10}+2a^{10}-2z^{7}a^{9}+9z^{5}a^{9}-16z^{3}a^{9}+12za^{9}-3a^{9}z^{-1}-2z^{8}a^{8}+8z^{6}a^{8}-9z^{4}a^{8}+4z^{2}a^{8}-z^{9}a^{7}+z^{7}a^{7}+12z^{5}a^{7}-25z^{3}a^{7}+17za^{7}-3a^{7}z^{-1}-4z^{8}a^{6}+19z^{6}a^{6}-25z^{4}a^{6}+12z^{2}a^{6}-2a^{6}-z^{9}a^{5}+2z^{7}a^{5}+8z^{5}a^{5}-18z^{3}a^{5}+11za^{5}-2a^{5}z^{-1}-2z^{8}a^{4}+10z^{6}a^{4}-13z^{4}a^{4}+4z^{2}a^{4}-z^{7}a^{3}+5z^{5}a^{3}-7z^{3}a^{3}+4za^{3}-a^{3}z^{-1}}$ (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 \
-7-6-5-4-3-2-1012χ
0         11
-2        1 -1
-4       21 1
-6     132  0
-8     31   2
-10   133    1
-12  143     0
-14  12      1
-16 23       -1
-18 1        1
-201         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-6}$ ${\displaystyle i=-4}$ ${\displaystyle i=-2}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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