# L11n248

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n248 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1) \left(u^2 v^2-u^2 v+u^2-u v^2+3 u v-u+v^2-v+1\right)}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $-6 q^{9/2}+10 q^{7/2}-\frac{1}{q^{7/2}}-14 q^{5/2}+\frac{4}{q^{5/2}}+15 q^{3/2}-\frac{9}{q^{3/2}}+2 q^{11/2}-15 \sqrt{q}+\frac{12}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z a^{-5} +z^5 a^{-3} +z^3 a^{-3} -z^7 a^{-1} +a z^5-4 z^5 a^{-1} +2 a z^3-6 z^3 a^{-1} +2 a z-3 z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $3 z^4 a^{-6} -3 z^2 a^{-6} +z^7 a^{-5} +5 z^5 a^{-5} -7 z^3 a^{-5} +2 z a^{-5} +3 z^8 a^{-4} -z^6 a^{-4} +z^4 a^{-4} -z^2 a^{-4} +2 z^9 a^{-3} +4 z^7 a^{-3} +a^3 z^5-7 z^5 a^{-3} -a^3 z^3+2 z^3 a^{-3} +10 z^8 a^{-2} +4 a^2 z^6-14 z^6 a^{-2} -5 a^2 z^4+3 z^4 a^{-2} +a^2 z^2+2 z^2 a^{-2} +2 z^9 a^{-1} +8 a z^7+11 z^7 a^{-1} -15 a z^5-28 z^5 a^{-1} +9 a z^3+19 z^3 a^{-1} -4 a z-6 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +7 z^8-9 z^6+z^2-1$ (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
12         2-2
10        4 4
8       62 -4
6      84  4
4     76   -1
2    88    0
0   69     3
-2  36      -3
-4 16       5
-6 3        -3
-81         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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