# L11a289

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a289 at Knotilus!
 Celtic or pseudo-Celtic linear decorative knot Decorative variant with big loops at ends

 Planar diagram presentation X10,1,11,2 X12,4,13,3 X22,12,9,11 X2,9,3,10 X20,14,21,13 X14,7,15,8 X18,16,19,15 X16,6,17,5 X6,18,7,17 X4,19,5,20 X8,22,1,21 Gauss code {1, -4, 2, -10, 8, -9, 6, -11}, {4, -1, 3, -2, 5, -6, 7, -8, 9, -7, 10, -5, 11, -3}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(u-1)(v-1)\left(u^{2}v^{2}-2u^{2}v+u^{2}-2uv^{2}+2uv-2u+v^{2}-2v+1\right)}{u^{3/2}v^{3/2}}}}$ (db) Jones polynomial ${\displaystyle 16q^{9/2}-18q^{7/2}+17q^{5/2}-{\frac {1}{q^{5/2}}}-15q^{3/2}+{\frac {3}{q^{3/2}}}+q^{17/2}-4q^{15/2}+8q^{13/2}-12q^{11/2}+10{\sqrt {q}}-{\frac {7}{\sqrt {q}}}}$ (db) Signature 3 (db) HOMFLY-PT polynomial ${\displaystyle z^{3}a^{-7}+za^{-7}-2z^{5}a^{-5}-5z^{3}a^{-5}-3za^{-5}+z^{7}a^{-3}+4z^{5}a^{-3}+7z^{3}a^{-3}+5za^{-3}-2z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+2az-5za^{-1}+az^{-1}-a^{-1}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle z^{4}a^{-10}+4z^{5}a^{-9}-2z^{3}a^{-9}+8z^{6}a^{-8}-8z^{4}a^{-8}+3z^{2}a^{-8}+10z^{7}a^{-7}-12z^{5}a^{-7}+6z^{3}a^{-7}-2za^{-7}+8z^{8}a^{-6}-4z^{6}a^{-6}-9z^{4}a^{-6}+4z^{2}a^{-6}+4z^{9}a^{-5}+9z^{7}a^{-5}-34z^{5}a^{-5}+24z^{3}a^{-5}-6za^{-5}+z^{10}a^{-4}+13z^{8}a^{-4}-33z^{6}a^{-4}+16z^{4}a^{-4}+7z^{9}a^{-3}-7z^{7}a^{-3}-24z^{5}a^{-3}+32z^{3}a^{-3}-10za^{-3}+z^{10}a^{-2}+8z^{8}a^{-2}-32z^{6}a^{-2}+28z^{4}a^{-2}-4z^{2}a^{-2}+3z^{9}a^{-1}+az^{7}-5z^{7}a^{-1}-4az^{5}-10z^{5}a^{-1}+6az^{3}+22z^{3}a^{-1}-4az-10za^{-1}+az^{-1}+a^{-1}z^{-1}+3z^{8}-11z^{6}+12z^{4}-3z^{2}-1}$ (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 \
-4-3-2-101234567χ
18           1-1
16          3 3
14         51 -4
12        73  4
10       95   -4
8      97    2
6     89     1
4    79      -2
2   510       5
0  25        -3
-2 15         4
-4 2          -2
-61           1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=2}$ ${\displaystyle i=4}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}}$ ${\displaystyle {\mathbb {Z} }^{9}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}}$ ${\displaystyle {\mathbb {Z} }^{9}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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