# L9n18

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n18 at Knotilus! L9n18 is ${\displaystyle 9_{53}^{2}}$ in the Rolfsen table of links.

 Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,5,9,6 X7,14,8,15 X13,16,14,17 X15,8,16,1 X6,9,7,10 X4,17,5,18 Gauss code {1, -2, 3, -9, 4, -8, -5, 7}, {8, -1, 2, -3, -6, 5, -7, 6, 9, -4}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {\left(t(2)t(1)^{2}+1\right)\left(t(1)t(2)^{2}+1\right)}{t(1)^{3/2}t(2)^{3/2}}}}$ (db) Jones polynomial ${\displaystyle -{\frac {1}{q^{7/2}}}-{\frac {1}{q^{11/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{21/2}}}}$ (db) Signature -6 (db) HOMFLY-PT polynomial ${\displaystyle a^{11}(-z)+a^{9}z^{5}+6a^{9}z^{3}+8a^{9}z+a^{9}z^{-1}-a^{7}z^{7}-7a^{7}z^{5}-15a^{7}z^{3}-11a^{7}z-a^{7}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle -z^{3}a^{13}+3za^{13}+za^{11}-z^{6}a^{10}+6z^{4}a^{10}-8z^{2}a^{10}-z^{7}a^{9}+7z^{5}a^{9}-14z^{3}a^{9}+9za^{9}-a^{9}z^{-1}-z^{6}a^{8}+6z^{4}a^{8}-8z^{2}a^{8}+a^{8}-z^{7}a^{7}+7z^{5}a^{7}-15z^{3}a^{7}+11za^{7}-a^{7}z^{-1}}$ (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 \
-8-7-6-5-4-3-2-10χ
-6        11
-8        11
-10      1  1
-12    1    1
-14   111   -1
-16   1     -1
-18  11     0
-201        1
-221        1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-8}$ ${\displaystyle i=-6}$ ${\displaystyle i=-4}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-7}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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