# L11a384

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a384 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^4 v^2-u^4 v+3 u^3 v^3-8 u^3 v^2+5 u^3 v-u^3+u^2 v^4-8 u^2 v^3+13 u^2 v^2-8 u^2 v+u^2-u v^4+5 u v^3-8 u v^2+3 u v-v^3+v^2}{u^2 v^2}$ (db) Jones polynomial $22 q^{9/2}-19 q^{7/2}+14 q^{5/2}-8 q^{3/2}+q^{21/2}-4 q^{19/2}+9 q^{17/2}-15 q^{15/2}+19 q^{13/2}-23 q^{11/2}+3 \sqrt{q}-\frac{1}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^3 a^{-9} -z^5 a^{-7} +z^3 a^{-7} +2 z a^{-7} - a^{-7} z^{-1} -3 z^5 a^{-5} -6 z^3 a^{-5} -3 z a^{-5} + a^{-5} z^{-1} -z^5 a^{-3} +z^3 a^{-3} +3 z a^{-3} +z^3 a^{-1} +z a^{-1}$ (db) Kauffman polynomial $-2 z^{10} a^{-6} -2 z^{10} a^{-8} -6 z^9 a^{-5} -12 z^9 a^{-7} -6 z^9 a^{-9} -8 z^8 a^{-4} -13 z^8 a^{-6} -12 z^8 a^{-8} -7 z^8 a^{-10} -6 z^7 a^{-3} +2 z^7 a^{-5} +18 z^7 a^{-7} +6 z^7 a^{-9} -4 z^7 a^{-11} -3 z^6 a^{-2} +13 z^6 a^{-4} +33 z^6 a^{-6} +33 z^6 a^{-8} +15 z^6 a^{-10} -z^6 a^{-12} -z^5 a^{-1} +9 z^5 a^{-3} +8 z^5 a^{-5} -4 z^5 a^{-7} +7 z^5 a^{-9} +9 z^5 a^{-11} +4 z^4 a^{-2} -13 z^4 a^{-4} -28 z^4 a^{-6} -21 z^4 a^{-8} -8 z^4 a^{-10} +2 z^4 a^{-12} +2 z^3 a^{-1} -7 z^3 a^{-3} -8 z^3 a^{-5} +2 z^3 a^{-7} -5 z^3 a^{-9} -6 z^3 a^{-11} -z^2 a^{-2} +6 z^2 a^{-4} +10 z^2 a^{-6} +5 z^2 a^{-8} +z^2 a^{-10} -z^2 a^{-12} -z a^{-1} +3 z a^{-3} -z a^{-5} -5 z a^{-7} +z a^{-9} +z a^{-11} - a^{-6} + a^{-5} z^{-1} + a^{-7} z^{-1}$ (db)

### Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).
 \ r \ j \
-2-10123456789χ
22           1-1
20          3 3
18         61 -5
16        93  6
14       117   -4
12      128    4
10     1011     1
8    912      -3
6   510       5
4  39        -6
2 16         5
0 2          -2
-21           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=4$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=5$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=6$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{9}$ $r=7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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