# L9a9

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a9 at Knotilus! L9a9 is ${\displaystyle 9_{37}^{2}}$ in the Rolfsen table of links.

 Planar diagram presentation X6172 X14,7,15,8 X16,9,17,10 X8,15,9,16 X4,17,1,18 X12,6,13,5 X10,4,11,3 X18,12,5,11 X2,14,3,13 Gauss code {1, -9, 7, -5}, {6, -1, 2, -4, 3, -7, 8, -6, 9, -2, 4, -3, 5, -8}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}+1\right)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}}$ (db) Jones polynomial ${\displaystyle -{\frac {3}{q^{9/2}}}-q^{7/2}+{\frac {5}{q^{7/2}}}+3q^{5/2}-{\frac {7}{q^{5/2}}}-5q^{3/2}+{\frac {8}{q^{3/2}}}+{\frac {1}{q^{11/2}}}+6{\sqrt {q}}-{\frac {9}{\sqrt {q}}}}$ (db) Signature -1 (db) HOMFLY-PT polynomial ${\displaystyle -a^{3}z^{5}-3a^{3}z^{3}-2a^{3}z+az^{7}+5az^{5}-z^{5}a^{-1}+8az^{3}-3z^{3}a^{-1}+4az-2za^{-1}+az^{-1}-a^{-1}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle -2a^{2}z^{8}-2z^{8}-4a^{3}z^{7}-8az^{7}-4z^{7}a^{-1}-4a^{4}z^{6}-3z^{6}a^{-2}+z^{6}-3a^{5}z^{5}+7a^{3}z^{5}+22az^{5}+11z^{5}a^{-1}-z^{5}a^{-3}-a^{6}z^{4}+5a^{4}z^{4}+3a^{2}z^{4}+7z^{4}a^{-2}+4z^{4}+4a^{5}z^{3}-8a^{3}z^{3}-24az^{3}-10z^{3}a^{-1}+2z^{3}a^{-3}+a^{6}z^{2}-a^{4}z^{2}-3a^{2}z^{2}-2z^{2}a^{-2}-3z^{2}+4a^{3}z+8az+4za^{-1}+1-az^{-1}-a^{-1}z^{-1}}$ (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 \
-5-4-3-2-101234χ
8         11
6        2 -2
4       31 2
2      32  -1
0     63   3
-2    45    1
-4   34     -1
-6  24      2
-8 13       -2
-10 2        2
-121         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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