# L9a33

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a33 at Knotilus! L9a33 is ${\displaystyle 9_{24}^{2}}$ in the Rolfsen table of links.

 Symmetric form Alternate symmetric version, with three lines touching at center Alternate symmetric version, with three lines touching at circumference Form made from 45-degree lines and circular arcs. Depiction obtained by knotilus. With an hypotrochoid [1]. Mexican book.

 Planar diagram presentation X8192 X12,3,13,4 X18,10,7,9 X10,14,11,13 X16,5,17,6 X14,18,15,17 X2738 X4,11,5,12 X6,15,1,16 Gauss code {1, -7, 2, -8, 5, -9}, {7, -1, 3, -4, 8, -2, 4, -6, 9, -5, 6, -3}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {u^{2}v^{2}-3u^{2}v+3u^{2}-3uv^{2}+7uv-3u+3v^{2}-3v+1}{uv}}}$ (db) Jones polynomial ${\displaystyle -{\frac {6}{q^{9/2}}}+{\frac {7}{q^{7/2}}}+q^{5/2}-{\frac {9}{q^{5/2}}}-4q^{3/2}+{\frac {10}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+6{\sqrt {q}}-{\frac {8}{\sqrt {q}}}}$ (db) Signature -1 (db) HOMFLY-PT polynomial ${\displaystyle a^{7}z^{-1}-3a^{5}z-a^{5}z^{-1}+3a^{3}z^{3}+3a^{3}z-az^{5}-2az^{3}+z^{3}a^{-1}-3az}$ (db) Kauffman polynomial ${\displaystyle -a^{4}z^{8}-a^{2}z^{8}-3a^{5}z^{7}-7a^{3}z^{7}-4az^{7}-2a^{6}z^{6}-7a^{4}z^{6}-11a^{2}z^{6}-6z^{6}-a^{7}z^{5}+5a^{5}z^{5}+9a^{3}z^{5}-az^{5}-4z^{5}a^{-1}+3a^{6}z^{4}+18a^{4}z^{4}+24a^{2}z^{4}-z^{4}a^{-2}+8z^{4}+3a^{7}z^{3}-3a^{5}z^{3}-2a^{3}z^{3}+8az^{3}+4z^{3}a^{-1}-11a^{4}z^{2}-14a^{2}z^{2}-3z^{2}-3a^{7}z+a^{5}z+2a^{3}z-2az-a^{6}+a^{7}z^{-1}+a^{5}z^{-1}}$ (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 \
-6-5-4-3-2-10123χ
6         1-1
4        3 3
2       31 -2
0      53  2
-2     64   -2
-4    34    -1
-6   46     2
-8  23      -1
-10  4       4
-1212        -1
-141         1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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