# L11n99

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n99 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v^3-u v^2-u v+2 u+2 v^5-v^4-v^3+v^2}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $q^{15/2}-q^{13/2}+q^{11/2}-2 q^{3/2}+\sqrt{q}-\frac{2}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $-z^5 a^{-1} -z^5 a^{-3} +a z^3-4 z^3 a^{-1} -5 z^3 a^{-3} +3 a z-2 z a^{-1} -6 z a^{-3} +2 z a^{-5} +z a^{-7} +a z^{-1} + a^{-1} z^{-1} -4 a^{-3} z^{-1} +2 a^{-5} z^{-1}$ (db) Kauffman polynomial $-z^9 a^{-1} -z^9 a^{-3} -2 z^8 a^{-2} -z^8 a^{-4} -z^8-a z^7+6 z^7 a^{-1} +8 z^7 a^{-3} -z^7 a^{-7} +13 z^6 a^{-2} +8 z^6 a^{-4} -z^6 a^{-6} -z^6 a^{-8} +5 z^6+6 a z^5-11 z^5 a^{-1} -22 z^5 a^{-3} +5 z^5 a^{-7} -25 z^4 a^{-2} -19 z^4 a^{-4} +6 z^4 a^{-6} +5 z^4 a^{-8} -5 z^4-10 a z^3+10 z^3 a^{-1} +29 z^3 a^{-3} +4 z^3 a^{-5} -5 z^3 a^{-7} +19 z^2 a^{-2} +19 z^2 a^{-4} -7 z^2 a^{-6} -6 z^2 a^{-8} -z^2+5 a z-5 z a^{-1} -16 z a^{-3} -6 z a^{-5} -5 a^{-2} -6 a^{-4} + a^{-6} +2 a^{-8} +1-a z^{-1} + a^{-1} z^{-1} +4 a^{-3} z^{-1} +2 a^{-5} z^{-1}$ (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 \
-4-3-2-101234567χ
16           1-1
14            0
12         11 0
10       21   -1
8       11   0
6     231    0
4    1 1     2
2   131      1
0  111       1
-2  1         1
-411          0
-61           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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