# L11n250

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n250 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^3 v^3-3 u^3 v^2+2 u^3 v-u^2 v^3+4 u^2 v^2-2 u^2 v-2 u v^2+4 u v-u+2 v^2-3 v+1}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $q^{13/2}-4 q^{11/2}+6 q^{9/2}-8 q^{7/2}+8 q^{5/2}-9 q^{3/2}+7 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^7 a^{-3} -2 z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} +a z^3-7 z^3 a^{-1} +9 z^3 a^{-3} -4 z^3 a^{-5} +2 a z-5 z a^{-1} +8 z a^{-3} -4 z a^{-5} +z a^{-7} + a^{-3} z^{-1} - a^{-5} z^{-1}$ (db) Kauffman polynomial $z^2 a^{-8} +4 z^3 a^{-7} -z a^{-7} +2 z^6 a^{-6} -z^4 a^{-6} +z^2 a^{-6} +5 z^7 a^{-5} -14 z^5 a^{-5} +14 z^3 a^{-5} -6 z a^{-5} + a^{-5} z^{-1} +5 z^8 a^{-4} -15 z^6 a^{-4} +12 z^4 a^{-4} -3 z^2 a^{-4} - a^{-4} +2 z^9 a^{-3} -17 z^5 a^{-3} +18 z^3 a^{-3} -7 z a^{-3} + a^{-3} z^{-1} +8 z^8 a^{-2} -30 z^6 a^{-2} +29 z^4 a^{-2} -8 z^2 a^{-2} +2 z^9 a^{-1} +a z^7-4 z^7 a^{-1} -4 a z^5-7 z^5 a^{-1} +5 a z^3+13 z^3 a^{-1} -2 a z-4 z a^{-1} +3 z^8-13 z^6+16 z^4-5 z^2$ (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 \
-4-3-2-1012345χ
14         1-1
12        3 3
10       42 -2
8      42  2
6     44   0
4    54    1
2   35     2
0  24      -2
-2 13       2
-4 2        -2
-61         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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