# L11n3

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n3 at Knotilus!

 Link L11n3. A graph, L11n3. 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 X8493 X11,20,12,21 X17,5,18,22 X21,19,22,18 X19,12,20,13 X9,16,10,17 X2,14,3,13 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 ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {(u-1)(v-1)\left(v^{2}-3v+1\right)}{{\sqrt {u}}v^{3/2}}}}$ (db) Jones polynomial ${\displaystyle -{\frac {6}{q^{9/2}}}+{\frac {6}{q^{7/2}}}-{\frac {7}{q^{5/2}}}-q^{3/2}+{\frac {6}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+2{\sqrt {q}}-{\frac {5}{\sqrt {q}}}}$ (db) Signature -1 (db) HOMFLY-PT polynomial ${\displaystyle a^{7}(-z)-a^{7}z^{-1}+2a^{5}z^{3}+4a^{5}z+3a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-5a^{3}z-3a^{3}z^{-1}+2az^{3}+3az+2az^{-1}-za^{-1}-a^{-1}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle -a^{5}z^{9}-a^{3}z^{9}-2a^{6}z^{8}-4a^{4}z^{8}-2a^{2}z^{8}-2a^{7}z^{7}+a^{3}z^{7}-az^{7}-a^{8}z^{6}+5a^{6}z^{6}+14a^{4}z^{6}+8a^{2}z^{6}+7a^{7}z^{5}+9a^{5}z^{5}+5a^{3}z^{5}+3az^{5}+4a^{8}z^{4}+a^{6}z^{4}-16a^{4}z^{4}-15a^{2}z^{4}-2z^{4}-6a^{7}z^{3}-13a^{5}z^{3}-15a^{3}z^{3}-9az^{3}-z^{3}a^{-1}-4a^{8}z^{2}-4a^{6}z^{2}+6a^{4}z^{2}+8a^{2}z^{2}+2z^{2}+3a^{7}z+10a^{5}z+12a^{3}z+7az+2za^{-1}+a^{8}+2a^{6}-2a^{2}-a^{7}z^{-1}-3a^{5}z^{-1}-3a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}}$ (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
4         11
2        1 -1
0       41 3
-2      43  -1
-4     32   1
-6    34    1
-8   33     0
-10  13      2
-12 13       -2
-14 1        1
-161         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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