# L11n206

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n206 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)-1) (t(1) t(2)+1)^2}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-4 q^{9/2}+5 q^{7/2}-6 q^{5/2}+4 q^{3/2}-\frac{1}{q^{3/2}}+q^{15/2}-2 q^{13/2}+3 q^{11/2}-4 \sqrt{q}+\frac{2}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z^5 a^{-5} +4 z^3 a^{-5} +4 z a^{-5} + a^{-5} z^{-1} -z^7 a^{-3} -6 z^5 a^{-3} -12 z^3 a^{-3} -9 z a^{-3} -3 a^{-3} z^{-1} +z^5 a^{-1} +4 z^3 a^{-1} +5 z a^{-1} +2 a^{-1} z^{-1}$ (db) Kauffman polynomial $z^6 a^{-8} -4 z^4 a^{-8} +3 z^2 a^{-8} +2 z^7 a^{-7} -8 z^5 a^{-7} +7 z^3 a^{-7} -z a^{-7} +2 z^8 a^{-6} -8 z^6 a^{-6} +9 z^4 a^{-6} -5 z^2 a^{-6} + a^{-6} +z^9 a^{-5} -3 z^7 a^{-5} +3 z^5 a^{-5} -5 z^3 a^{-5} +4 z a^{-5} - a^{-5} z^{-1} +3 z^8 a^{-4} -13 z^6 a^{-4} +21 z^4 a^{-4} -14 z^2 a^{-4} +3 a^{-4} +z^9 a^{-3} -5 z^7 a^{-3} +14 z^5 a^{-3} -18 z^3 a^{-3} +11 z a^{-3} -3 a^{-3} z^{-1} +z^8 a^{-2} -4 z^6 a^{-2} +10 z^4 a^{-2} -8 z^2 a^{-2} +3 a^{-2} +3 z^5 a^{-1} +a z^3-5 z^3 a^{-1} -a z+5 z a^{-1} -2 a^{-1} z^{-1} +2 z^4-2 z^2$ (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 \
-2-101234567χ
16         1-1
14        1 1
12       21 -1
10      21  1
8     32   -1
6    32    1
4   13     2
2  33      0
0 13       2
-2 1        -1
-41         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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