# L11n164

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n164 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^2 v^3-u^2 v^2+2 u^2 v-u^2+u v^4-2 u v^3+3 u v^2-2 u v+u-v^4+2 v^3-v^2+v}{u v^2}$ (db) Jones polynomial $-\frac{6}{q^{9/2}}+\frac{6}{q^{7/2}}-\frac{6}{q^{5/2}}+\frac{4}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{5}{q^{11/2}}+\sqrt{q}-\frac{3}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 z^3+2 a^7 z+a^7 z^{-1} -a^5 z^5-3 a^5 z^3-3 a^5 z-a^5 z^{-1} -a^3 z^5-3 a^3 z^3-3 a^3 z+a z^3+a z$ (db) Kauffman polynomial $a^9 z^7-5 a^9 z^5+8 a^9 z^3-4 a^9 z+2 a^8 z^8-9 a^8 z^6+11 a^8 z^4-3 a^8 z^2+a^7 z^9-a^7 z^7-8 a^7 z^5+10 a^7 z^3-a^7 z^{-1} +4 a^6 z^8-14 a^6 z^6+11 a^6 z^4-3 a^6 z^2+a^6+a^5 z^9-7 a^5 z^5+3 a^5 z^3+2 a^5 z-a^5 z^{-1} +2 a^4 z^8-4 a^4 z^6+2 a^3 z^7-4 a^3 z^5+4 a^3 z^3-3 a^3 z+a^2 z^6+a^2 z^2+3 a z^3-a z+z^2$ (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 \
-8-7-6-5-4-3-2-101χ
2         1-1
0        2 2
-2       32 -1
-4      31  2
-6     33   0
-8    33    0
-10   23     1
-12  23      -1
-14 13       2
-16 1        -1
-181         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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