# L11n450

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n450 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(w-1) (x-1)^2 (u v x+u (-v)+u-v x+x-1)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}}$ (db) Jones polynomial $q^{9/2}-3 q^{7/2}+7 q^{5/2}-13 q^{3/2}+14 \sqrt{q}-\frac{18}{\sqrt{q}}+\frac{14}{q^{3/2}}-\frac{14}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{4}{q^{9/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a z^7-a^3 z^5+4 a z^5-2 z^5 a^{-1} -3 a^3 z^3+8 a z^3-6 z^3 a^{-1} +z^3 a^{-3} +a^5 z-7 a^3 z+12 a z-8 z a^{-1} +2 z a^{-3} +2 a^5 z^{-1} -8 a^3 z^{-1} +11 a z^{-1} -6 a^{-1} z^{-1} + a^{-3} z^{-1} +a^5 z^{-3} -3 a^3 z^{-3} +3 a z^{-3} - a^{-1} z^{-3}$ (db) Kauffman polynomial $10 a^5 z^3-a^5 z^{-3} -11 a^5 z+5 a^5 z^{-1} +6 a^4 z^6+z^6 a^{-4} -4 a^4 z^4-3 z^4 a^{-4} +9 a^4 z^2+3 z^2 a^{-4} +3 a^4 z^{-2} -9 a^4- a^{-4} +13 a^3 z^7+3 z^7 a^{-3} -31 a^3 z^5-8 z^5 a^{-3} +44 a^3 z^3+9 z^3 a^{-3} -3 a^3 z^{-3} -34 a^3 z-6 z a^{-3} +14 a^3 z^{-1} +2 a^{-3} z^{-1} +9 a^2 z^8+4 z^8 a^{-2} -9 a^2 z^6-5 z^6 a^{-2} -9 a^2 z^4-7 z^4 a^{-2} +24 a^2 z^2+14 z^2 a^{-2} +6 a^2 z^{-2} -21 a^2-6 a^{-2} +2 a z^9+2 z^9 a^{-1} +20 a z^7+10 z^7 a^{-1} -62 a z^5-39 z^5 a^{-1} +67 a z^3+42 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -44 a z-27 z a^{-1} +18 a z^{-1} +11 a^{-1} z^{-1} +13 z^8-21 z^6-9 z^4+26 z^2+3 z^{-2} -18$ (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        2 2
6       51 -4
4      82  6
2     65   -1
0    128    4
-2   812     4
-4  66      0
-6 28       6
-826        -4
-104         4
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}^{4}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{12}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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