# L11n88

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n88 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1)^3 \left(v^2-v+1\right)}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $q^{9/2}-\frac{3}{q^{9/2}}-4 q^{7/2}+\frac{8}{q^{7/2}}+8 q^{5/2}-\frac{12}{q^{5/2}}-13 q^{3/2}+\frac{15}{q^{3/2}}+15 \sqrt{q}-\frac{17}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a z^7-a^3 z^5+4 a z^5-2 z^5 a^{-1} -3 a^3 z^3+7 a z^3-5 z^3 a^{-1} +z^3 a^{-3} +a^5 z-5 a^3 z+7 a z-4 z a^{-1} +z a^{-3} +a^5 z^{-1} -3 a^3 z^{-1} +4 a z^{-1} -2 a^{-1} z^{-1}$ (db) Kauffman polynomial $6 a^5 z^3-4 a^5 z+a^5 z^{-1} +3 a^4 z^6+z^6 a^{-4} +6 a^4 z^4-2 z^4 a^{-4} -7 a^4 z^2+z^2 a^{-4} +a^4+10 a^3 z^7+4 z^7 a^{-3} -21 a^3 z^5-10 z^5 a^{-3} +28 a^3 z^3+8 z^3 a^{-3} -16 a^3 z-3 z a^{-3} +3 a^3 z^{-1} +10 a^2 z^8+6 z^8 a^{-2} -22 a^2 z^6-13 z^6 a^{-2} +27 a^2 z^4+5 z^4 a^{-2} -17 a^2 z^2+3 a^2+ a^{-2} +3 a z^9+3 z^9 a^{-1} +13 a z^7+7 z^7 a^{-1} -48 a z^5-37 z^5 a^{-1} +49 a z^3+35 z^3 a^{-1} -22 a z-13 z a^{-1} +4 a z^{-1} +2 a^{-1} z^{-1} +16 z^8-39 z^6+28 z^4-11 z^2+2$ (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 \
-4-3-2-1012345χ
10         1-1
8        3 3
6       51 -4
4      83  5
2     75   -2
0    108    2
-2   79     2
-4  58      -3
-6 37       4
-8 5        -5
-103         3
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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