# L9a32

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a32 at Knotilus! L9a32 is ${\displaystyle 9_{40}^{2}}$ in the Rolfsen table of links.
 A traditional symbol of the Christian Trinity (a triquetra interlaced with a circle, or "Trinity knot") Symmetric version, with three lines touching at center Symmetric version, with three lines touching at cicrumference

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

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {-t(2)^{4}-3t(1)t(2)^{3}+3t(2)^{3}-3t(1)^{2}t(2)^{2}+5t(1)t(2)^{2}-3t(2)^{2}+3t(1)^{2}t(2)-3t(1)t(2)-t(1)^{2}}{t(1)t(2)^{2}}}}$ (db) Jones polynomial ${\displaystyle {\frac {7}{q^{9/2}}}-{\frac {6}{q^{7/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {5}{q^{17/2}}}-{\frac {7}{q^{15/2}}}+{\frac {8}{q^{13/2}}}-{\frac {9}{q^{11/2}}}}$ (db) Signature -3 (db) HOMFLY-PT polynomial ${\displaystyle -a^{11}z^{-1}+4a^{9}z+3a^{9}z^{-1}-3a^{7}z^{3}-4a^{7}z-2a^{7}z^{-1}-3a^{5}z^{3}-3a^{5}z-a^{3}z^{3}}$ (db) Kauffman polynomial ${\displaystyle a^{12}z^{6}-3a^{12}z^{4}+3a^{12}z^{2}-a^{12}+3a^{11}z^{7}-10a^{11}z^{5}+9a^{11}z^{3}-2a^{11}z+a^{11}z^{-1}+2a^{10}z^{8}-a^{10}z^{6}-12a^{10}z^{4}+14a^{10}z^{2}-3a^{10}+9a^{9}z^{7}-26a^{9}z^{5}+22a^{9}z^{3}-11a^{9}z+3a^{9}z^{-1}+2a^{8}z^{8}+5a^{8}z^{6}-21a^{8}z^{4}+14a^{8}z^{2}-3a^{8}+6a^{7}z^{7}-10a^{7}z^{5}+6a^{7}z^{3}-6a^{7}z+2a^{7}z^{-1}+7a^{6}z^{6}-9a^{6}z^{4}+3a^{6}z^{2}+6a^{5}z^{5}-6a^{5}z^{3}+3a^{5}z+3a^{4}z^{4}+a^{3}z^{3}}$ (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 \
-9-8-7-6-5-4-3-2-10χ
-2         11
-4        31-2
-6       3  3
-8      43  -1
-10     53   2
-12    34    1
-14   45     -1
-16  24      2
-18 13       -2
-20 2        2
-221         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-4}$ ${\displaystyle i=-2}$ ${\displaystyle r=-9}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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