# L9a18

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a18 at Knotilus! L9a18 is is ${\displaystyle 9_{10}^{2}}$ in the Rolfsen table of links.

 Planar diagram presentation X6172 X12,4,13,3 X18,8,5,7 X16,10,17,9 X14,12,15,11 X10,16,11,15 X8,18,9,17 X2536 X4,14,1,13 Gauss code {1, -8, 2, -9}, {8, -1, 3, -7, 4, -6, 5, -2, 9, -5, 6, -4, 7, -3}
A Braid Representative
A Morse Link Presentation

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {3(t(1)-1)(t(2)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}}}}$ (db) Jones polynomial ${\displaystyle -3q^{9/2}+4q^{7/2}-4q^{5/2}+3q^{3/2}-{\frac {1}{q^{3/2}}}+q^{15/2}-2q^{13/2}+2q^{11/2}-3{\sqrt {q}}+{\frac {1}{\sqrt {q}}}}$ (db) Signature 1 (db) HOMFLY-PT polynomial ${\displaystyle za^{-7}-z^{3}a^{-5}-za^{-5}-z^{3}a^{-3}-z^{3}a^{-1}+az-za^{-1}+az^{-1}-a^{-1}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle -z^{8}a^{-4}-z^{8}a^{-6}-z^{7}a^{-3}-3z^{7}a^{-5}-2z^{7}a^{-7}-z^{6}a^{-2}+3z^{6}a^{-4}+3z^{6}a^{-6}-z^{6}a^{-8}-z^{5}a^{-1}+z^{5}a^{-3}+11z^{5}a^{-5}+9z^{5}a^{-7}-4z^{4}a^{-4}-z^{4}a^{-6}+4z^{4}a^{-8}-z^{4}-az^{3}-z^{3}a^{-1}-10z^{3}a^{-5}-10z^{3}a^{-7}+3z^{2}a^{-4}-3z^{2}a^{-8}+2az+2za^{-1}+2za^{-5}+2za^{-7}+1-az^{-1}-a^{-1}z^{-1}}$ (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 \
-2-101234567χ
16         1-1
14        1 1
12       11 0
10      21  1
8     21   -1
6    22    0
4   12     1
2  22      0
0 13       2
-2          0
-41         1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=0}$ ${\displaystyle i=2}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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