# L11n249

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n249 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^3 v^3-3 u^3 v^2+2 u^3 v-2 u^2 v^3+7 u^2 v^2-7 u^2 v+u^2+u v^3-7 u v^2+7 u v-2 u+2 v^2-3 v+1}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $q^{9/2}-\frac{3}{q^{9/2}}-4 q^{7/2}+\frac{7}{q^{7/2}}+8 q^{5/2}-\frac{12}{q^{5/2}}-12 q^{3/2}+\frac{14}{q^{3/2}}+15 \sqrt{q}-\frac{16}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z+a^5 z^{-1} -a^3 z^5-3 a^3 z^3+z^3 a^{-3} -4 a^3 z+z a^{-3} -a^3 z^{-1} +a z^7+4 a z^5-2 z^5 a^{-1} +6 a z^3-5 z^3 a^{-1} +3 a z-3 z a^{-1}$ (db) Kauffman polynomial $-3 a z^9-3 z^9 a^{-1} -8 a^2 z^8-6 z^8 a^{-2} -14 z^8-8 a^3 z^7-8 a z^7-4 z^7 a^{-1} -4 z^7 a^{-3} -3 a^4 z^6+13 a^2 z^6+14 z^6 a^{-2} -z^6 a^{-4} +31 z^6+13 a^3 z^5+27 a z^5+24 z^5 a^{-1} +10 z^5 a^{-3} -3 a^4 z^4-10 a^2 z^4-7 z^4 a^{-2} +2 z^4 a^{-4} -16 z^4-6 a^5 z^3-15 a^3 z^3-18 a z^3-16 z^3 a^{-1} -7 z^3 a^{-3} +a^4 z^2+3 a^2 z^2-z^2 a^{-4} +3 z^2+5 a^5 z+7 a^3 z+4 a z+3 z a^{-1} +z a^{-3} +a^4-a^5 z^{-1} -a^3 z^{-1}$ (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 \
-4-3-2-1012345χ
10         1-1
8        3 3
6       51 -4
4      73  4
2     85   -3
0    87    1
-2   79     2
-4  57      -2
-6 27       5
-815        -4
-103         3
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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