# L11a114

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a114 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{4 (u-1) (v-1) \left(v^2-v+1\right)}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $15 q^{9/2}-15 q^{7/2}+10 q^{5/2}-7 q^{3/2}+q^{21/2}-2 q^{19/2}+5 q^{17/2}-9 q^{15/2}+13 q^{13/2}-15 q^{11/2}+3 \sqrt{q}-\frac{1}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^3 a^{-9} +2 z a^{-9} + a^{-9} z^{-1} -z^5 a^{-7} -2 z^3 a^{-7} -3 z a^{-7} - a^{-7} z^{-1} -2 z^5 a^{-5} -4 z^3 a^{-5} -3 z a^{-5} -2 a^{-5} z^{-1} -z^5 a^{-3} +3 z a^{-3} +2 a^{-3} z^{-1} +z^3 a^{-1} +z a^{-1}$ (db) Kauffman polynomial $z^6 a^{-12} -4 z^4 a^{-12} +4 z^2 a^{-12} +2 z^7 a^{-11} -6 z^5 a^{-11} +4 z^3 a^{-11} +3 z^8 a^{-10} -9 z^6 a^{-10} +12 z^4 a^{-10} -12 z^2 a^{-10} +4 a^{-10} +3 z^9 a^{-9} -8 z^7 a^{-9} +12 z^5 a^{-9} -11 z^3 a^{-9} +2 z a^{-9} - a^{-9} z^{-1} +z^{10} a^{-8} +5 z^8 a^{-8} -22 z^6 a^{-8} +39 z^4 a^{-8} -34 z^2 a^{-8} +9 a^{-8} +6 z^9 a^{-7} -12 z^7 a^{-7} +8 z^5 a^{-7} +3 z^3 a^{-7} -z a^{-7} - a^{-7} z^{-1} +z^{10} a^{-6} +7 z^8 a^{-6} -21 z^6 a^{-6} +27 z^4 a^{-6} -15 z^2 a^{-6} +4 a^{-6} +3 z^9 a^{-5} +3 z^7 a^{-5} -20 z^5 a^{-5} +27 z^3 a^{-5} -11 z a^{-5} +2 a^{-5} z^{-1} +5 z^8 a^{-4} -6 z^6 a^{-4} -z^4 a^{-4} +4 z^2 a^{-4} -2 a^{-4} +5 z^7 a^{-3} -9 z^5 a^{-3} +7 z^3 a^{-3} -7 z a^{-3} +2 a^{-3} z^{-1} +3 z^6 a^{-2} -5 z^4 a^{-2} +z^2 a^{-2} +z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1}$ (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          1 1
18         41 -3
16        51  4
14       84   -4
12      75    2
10     88     0
8    77      0
6   38       5
4  47        -3
2 15         4
0 2          -2
-21           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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