# L11n234

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n234 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(1)^2 t(2)^2-t(1) t(2)^2+5 t(1) t(2)-t(1)+1\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{5}{q^{9/2}}-q^{7/2}+\frac{8}{q^{7/2}}+3 q^{5/2}-\frac{12}{q^{5/2}}-7 q^{3/2}+\frac{12}{q^{3/2}}+\frac{2}{q^{11/2}}+10 \sqrt{q}-\frac{12}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-a^5 z-a^3 z^5-2 a^3 z^3-2 a^3 z+a z^7+5 a z^5-z^5 a^{-1} +10 a z^3-3 z^3 a^{-1} +7 a z-4 z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-a^3 z^9-a z^9-2 a^4 z^8-6 a^2 z^8-4 z^8-a^5 z^7-5 a^3 z^7-9 a z^7-5 z^7 a^{-1} +6 a^2 z^6-3 z^6 a^{-2} +3 z^6-3 a^5 z^5+9 a^3 z^5+22 a z^5+9 z^5 a^{-1} -z^5 a^{-3} -3 a^6 z^4+a^2 z^4+5 z^4 a^{-2} +3 z^4+6 a^5 z^3-5 a^3 z^3-19 a z^3-6 z^3 a^{-1} +2 z^3 a^{-3} +4 a^6 z^2+a^4 z^2-2 a^2 z^2-2 z^2 a^{-2} -z^2-3 a^5 z+8 a z+4 z a^{-1} -z a^{-3} +1-a z^{-1} - a^{-1} z^{-1}$ (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 \
-5-4-3-2-101234χ
8         11
6        2 -2
4       51 4
2      52  -3
0     75   2
-2    77    0
-4   55     0
-6  37      4
-8 25       -3
-10 3        3
-122         -2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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