# L11n118

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n118 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-2) (v-1) (2 v-1)}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $-10 q^{9/2}+11 q^{7/2}-13 q^{5/2}+11 q^{3/2}-\frac{2}{q^{3/2}}+q^{15/2}-3 q^{13/2}+7 q^{11/2}-9 \sqrt{q}+\frac{5}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $2 z^5 a^{-3} -4 z^3 a^{-1} +6 z^3 a^{-3} -3 z^3 a^{-5} +2 a z-6 z a^{-1} +8 z a^{-3} -5 z a^{-5} +z a^{-7} +a z^{-1} -2 a^{-1} z^{-1} +3 a^{-3} z^{-1} -3 a^{-5} z^{-1} + a^{-7} z^{-1}$ (db) Kauffman polynomial $-2 z^9 a^{-3} -2 z^9 a^{-5} -6 z^8 a^{-2} -10 z^8 a^{-4} -4 z^8 a^{-6} -5 z^7 a^{-1} -6 z^7 a^{-3} -4 z^7 a^{-5} -3 z^7 a^{-7} +17 z^6 a^{-2} +27 z^6 a^{-4} +8 z^6 a^{-6} -z^6 a^{-8} -z^6+13 z^5 a^{-1} +30 z^5 a^{-3} +25 z^5 a^{-5} +8 z^5 a^{-7} -24 z^4 a^{-2} -22 z^4 a^{-4} +z^4 a^{-6} +3 z^4 a^{-8} -4 z^4-3 a z^3-23 z^3 a^{-1} -39 z^3 a^{-3} -25 z^3 a^{-5} -6 z^3 a^{-7} +11 z^2 a^{-2} +7 z^2 a^{-4} -3 z^2 a^{-6} -3 z^2 a^{-8} +4 z^2+4 a z+13 z a^{-1} +18 z a^{-3} +12 z a^{-5} +3 z a^{-7} -2 a^{-2} +2 a^{-6} + a^{-8} -a z^{-1} -2 a^{-1} z^{-1} -3 a^{-3} z^{-1} -3 a^{-5} z^{-1} - a^{-7} z^{-1}$ (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 \
-2-101234567χ
16         1-1
14        2 2
12       51 -4
10      52  3
8     65   -1
6    75    2
4   46     2
2  57      -2
0 26       4
-2 3        -3
-42         2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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