# L11n178

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n178 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{\left(u v^2-u v+u+v-1\right) \left(u v^2-u v-v^2+v-1\right)}{u v^2}$ (db) Jones polynomial $-q^{3/2}+3 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^3 z^7-a^5 z^5+5 a^3 z^5-a z^5-3 a^5 z^3+8 a^3 z^3-3 a z^3-2 a^5 z+4 a^3 z-3 a z+a^3 z^{-1} -a z^{-1}$ (db) Kauffman polynomial $-a^5 z^9-a^3 z^9-3 a^6 z^8-4 a^4 z^8-a^2 z^8-3 a^7 z^7-2 a^5 z^7+a^3 z^7-a^8 z^6+8 a^6 z^6+8 a^4 z^6-a^2 z^6+10 a^7 z^5+10 a^5 z^5-6 a^3 z^5-6 a z^5+3 a^8 z^4-3 a^6 z^4-3 a^4 z^4-3 z^4-8 a^7 z^3-4 a^5 z^3+12 a^3 z^3+7 a z^3-z^3 a^{-1} -2 a^8 z^2-a^6 z^2+a^4 z^2+a^2 z^2+z^2+a^7 z-a^5 z-6 a^3 z-4 a z-a^2+a^3 z^{-1} +a z^{-1}$ (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 \
-7-6-5-4-3-2-1012χ
4         11
2        2 -2
0       41 3
-2      43  -1
-4     53   2
-6    45    1
-8   34     -1
-10  24      2
-12 13       -2
-14 2        2
-161         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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